Journal of ISSN: 2377-4282JNMR

Nanomedicine Research
Research Article
Volume 4 Issue 3 - 2016
A New Nonrelativistic Investigation for Interactions in One-Electron Atoms With Modified Inverse-Square Potential: Noncommutative Two and Three Dimensional Space Phase Solutions at Planck’s and Nano-Scales
Abdelmadjid Maireche*
Department of Physics, University of M'sila M’sila, Algeria
Received: May 30, 2016 | Published: November 28, 2016
*Corresponding author: Abdelmadjid Maireche, Laboratory of Physics and Material Chemistry, Physics department, University of M'sila-M’sila Algeria, Tel: +213664834317; Email:
Citation: Maireche A (2016) A New Nonrelativistic Investigation for Interactions in One-Electron Atoms With Modified Inverse-Square Potential: Noncommutative Two and Three Dimensional Space Phase Solutions at Planck’s and Nano-Scales. J Nanomed Res 4(3): 00090. DOI: 10.15406/jnmr.2016.04.00090

Abstract

In this paper, we present a novel theoretical analytical perform further investigation for the exact solvability of non-relativistic quantum spectrum systems for modified inverse-square potential (m.i.s.) potential is discussed by means Boopp’s shift method instead to solving deformed Schrödinger equations with star product, in the framework of both noncommutativity (two -three) dimensional real space and phase (NC: 2D-RSP) and (NC: 3D-RSP). The exact corrections for excited states are found straightforwardly for interactions in one-electron atoms by means of the standard perturbation theory. Furthermore, the obtained corrections of energies are depended on four infinitesimals parameters ( , ) and ( , ), which are induced by position-position and momentum-momentum noncommutativity, (NC: 2D-RSP) and (NC: 3D-RSP), respectively, in addition to the discreet atomic quantum numbers: and (the angular momentum quantum number) and we have also shown that, the usual states in ordinary two and three dimensional spaces are cancelled and has been replaced by new degenerated sub-states in the new quantum symmetries of (NC: 2D-RSP) and (NC: 3D-RSP).

Keywords: The inverse-square potential; Noncommutative space; Phase; Star product; Boopp’s shift method.

Abbreviations

MIS: Modified Inverse Square potential; NC: 2D-3D-RSP: Noncommutativity (two-three) Dimensional Real Space Phase; CCRs: Canonical Commutations Relations; NNCCRs: New Noncommutative Canonical Commutations Relations; MSE: Modified Schrödinger Equations.

Introduction

It is well-known, that, the modern quantum mechanics, satisfied a big successful in the last few years, for describing atoms, nuclei, and molecules and their spectral behaviors based on three fundamental equations: Schrödinger, Klein-Gordon and Dirac. Schrödinger equation rest the first and the latest in terms of interest, it is playing a crucial role in devising well-behaved physical models in different fields of physics and chemists, many potentials are treated within the framework of nonrelativistic quantum mechanics based on this equation in two, three and D generalized spaces [1-32], the quantum structure based to the ordinary canonical commutations relations (CCRs) in both Schrödinger and Heisenberg (the operators are depended on time) pictures (CCRs), respectively, as:
[ x i , p j ]=i δ ij                  and    [ x i , x j ] = [ p i , p j ] = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba GaamiEamaaBaaajuaibaGaamyAaaqabaqcfaOaaiilaiaadchadaWg aaqcfasaaiaadQgaaKqbagqaaaGaay5waiaaw2faaiabg2da9iaadM gacqaH0oazdaWgaaqcfasaaiaadMgacaWGQbaajuaGbeaacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bggacaqGUbGaaeizaiaabccacaqGGaGaaeiiaiaabccadaWadaqaai aadIhadaWgaaqcfasaaiaadMgaaeqaaKqbakaacYcacaWG4bWaaSba aKqbGeaacaWGQbaajuaGbeaaaiaawUfacaGLDbaaqaaaaaaaaaWdbi aacckapaGaeyypa0ZdbiaacckapaWaamWaaeaacaWGWbWaaSbaaKqb GeaacaWGPbaabeaajuaGcaGGSaGaamiCamaaBaaajuaibaGaamOAaa qcfayabaaacaGLBbGaayzxaaWdbiaacckapaGaeyypa0Zdbiaaccka paGaaGimaaaa@6D4A@ (1.1)

[ x i ( t ), p j ( t ) ] = i δ ij            and    [ x i ( t ), x j ( t ) ] = [ p i ( t ), p j ( t ) ] = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba GaamiEamaaBaaajuaibaGaamyAaaqabaqcfa4aaeWaaeaacaWG0baa caGLOaGaayzkaaGaaiilaiaadchadaWgaaqcfasaaiaadQgaaKqbag qaamaabmaabaGaamiDaaGaayjkaiaawMcaaaGaay5waiaaw2faaaba aaaaaaaapeGaaiiOa8aacqGH9aqppeGaaiiOa8aacaWGPbGaeqiTdq 2aaSbaaKqbGeaacaWGPbGaamOAaaqabaqcfaOaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaaeiiaiaabccacaqGGaWaamWa aeaacaWG4bWaaSbaaKqbGeaacaWGPbaabeaajuaGdaqadaqaaiaads haaiaawIcacaGLPaaacaGGSaGaamiEamaaBaaajuaibaGaamOAaaqc fayabaWaaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaa WdbiaacckapaGaeyypa0ZdbiaacckapaWaamWaaeaacaWGWbWaaSba aKqbGeaacaWGPbaabeaajuaGdaqadaqaaiaadshaaiaawIcacaGLPa aacaGGSaGaamiCamaaBaaajuaibaGaamOAaaqcfayabaWaaeWaaeaa caWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaaWdbiaacckapaGaey ypa0ZdbiaacckapaGaaGimaaaa@7B0A@ (1.2)

Where the two operators ( x i ( t ), p i ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamiEamaaBaaajuaqbaGaamyAaaqabaqcfa4aaeWaaeaacaWG0baa caGLOaGaayzkaaGaaiilaiaadchadaWgaaqcfasaaiaadMgaaeqaaK qbaoaabmaabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa @435E@ in Heisenberg picture are related to the corresponding two operators ( x i , p i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamiEamaaBaaajuaibaGaamyAaaqabaqcfaOaaiilaiaadchadaWg aaqcfasaaiaadMgaaeqaaaqcfaOaayjkaiaawMcaaaaa@3E3A@ in Schrödinger picture from the two projections relations:

x i ( t )=exp(iH( t t 0 )) x i exp(iH( t t 0 ))    and     p i ( t )=exp(iH( t t 0 )) p i exp(iH( t t 0 )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadMgaaeqaaKqbaoaabmaabaGaamiDaaGaayjkaiaa wMcaaiabg2da9iGacwgacaGG4bGaaiiCaiaacIcacaWGPbGaamisam aabmaabaGaamiDaiabgkHiTiaadshadaWgaaqcfasaaiaaicdaaKqb agqaaaGaayjkaiaawMcaaiaacMcacaWG4bWaaSbaaKqbGeaacaWGPb aabeaajuaGciGGLbGaaiiEaiaacchacaGGOaGaeyOeI0IaamyAaiaa dIeadaqadaqaaiaadshacqGHsislcaWG0bWaaSbaaKqbGeaacaaIWa aajuaGbeaaaiaawIcacaGLPaaacaGGPaGaaeiiaiaabccacaqGGaGa aeiiaiaabggacaqGUbGaaeizaiaabccacaqGGaGaaeiiaiaabccaca WGWbWaaSbaaKqbGeaacaWGPbaabeaajuaGdaqadaqaaiaadshaaiaa wIcacaGLPaaacqGH9aqpciGGLbGaaiiEaiaacchacaGGOaGaamyAai aadIeadaqadaqaaiaadshacqGHsislcaWG0bWaaSbaaKqbGeaacaaI WaaajuaGbeaaaiaawIcacaGLPaaacaGGPaGaamiCamaaBaaajuaiba GaamyAaaqcfayabaGaciyzaiaacIhacaGGWbGaaiikaiabgkHiTiaa dMgacaWGibWaaeWaaeaacaWG0bGaeyOeI0IaamiDamaaBaaajuaiba GaaGimaaqcfayabaaacaGLOaGaayzkaaGaaiykaaaa@81FF@ (1.3)

Here denote to the ordinary quantum Hamiltonian operator, recently, much considerable effort has been expanded on the solutions of Schrödinger, Dirac and Klein-Gordon equations to noncommutative quantum mechanics, the present paper investigates first the present new quantum structure which based to new noncommutative canonical commutations relations (NNCCRs) in both Schrödinger and Heisenberg pictures, respectively, as follows [33-60]:

[ x ^ i , p ^ j ]=i δ ij ,[ x ^ i , x ^ j ]=i θ ij           and     [ p ^ i , p ^ j ]=i θ ¯ ij [ x ^ i ( t ) , p ^ j ( t ) ]=i δ ij ,[ x ^ i ( t ) , x ^ j ( t ) ]=i θ ij  and [ p ^ i ( t ) , p ^ j ( t ) ]=i θ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aam WaaeaaceWG4bGbaKaadaWgaaqcfasaaiaadMgaaeqaaKqbaoaaxaca baGaaiilaaqabKqbGeaacqGHxiIkaaqcfaOabmiCayaajaWaaSbaaK qbGeaacaWGQbaajuaGbeaaaiaawUfacaGLDbaacqGH9aqpcaWGPbGa eqiTdq2aaSbaaKqbGeaacaWGPbGaamOAaaqabaqcfaOaaiilamaadm aabaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGdaWfGaqa aiaacYcaaeqajuaibaGaey4fIOcaaKqbakqadIhagaqcamaaBaaaju aibaGaamOAaaqabaaajuaGcaGLBbGaayzxaaGaeyypa0JaamyAaiab eI7aXnaaBaaajuaibaGaamyAaiaadQgaaKqbagqaaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGHbGaaeOBaiaabsgacaqGGaGaaeiiaiaabccacaqGGaGaaeiiam aadmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGdaWf GaqaaiaacYcaaeqajuaibaGaey4fIOcaaKqbakqadchagaqcamaaBa aajuaibaGaamOAaaqcfayabaaacaGLBbGaayzxaaGaeyypa0JaamyA amaanaaabaGaeqiUdehaamaaBaaajuaibaGaamyAaiaadQgaaeqaaa Gcbaqcfa4aamWaaeaaceWG4bGbaKaadaWgaaqcfasaaiaadMgaaeqa aKqbaoaabmaabaGaamiDaaGaayjkaiaawMcaamaaxacabaGaaiilaa qabKqbGeaacqGHxiIkaaqcfaOabmiCayaajaWaaSbaaKqbGeaacaWG QbaabeaajuaGdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawUfaca GLDbaacqGH9aqpcaWGPbGaeqiTdq2aaSbaaKqbGeaacaWGPbGaamOA aaqabaqcfaOaaiilamaadmaabaGabmiEayaajaWaaSbaaKqbGeaaca WGPbaabeaajuaGdaqadaqaaiaadshaaiaawIcacaGLPaaadaWfGaqa aiaacYcaaeqajuaibaGaey4fIOcaaKqbakqadIhagaqcamaaBaaajq wba+FaaiaadQgaaKqbagqaamaabmaabaGaamiDaaGaayjkaiaawMca aaGaay5waiaaw2faaiabg2da9iaadMgacqaH4oqCdaWgaaqcfasaai aadMgacaWGQbaajuaGbeaacaqGGaGaaeyyaiaab6gacaqGKbGaaeii amaadmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGda qadaqaaiaadshaaiaawIcacaGLPaaadaWfGaqaaiaacYcaaeqajuai baGaey4fIOcaaKqbakqadchagaqcamaaBaaajuaibaGaamOAaaqcfa yabaWaaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaaGa eyypa0JaamyAamaanaaabaGaeqiUdehaamaaBaaajuaibaGaamyAai aadQgaaeqaaaaaaa@BC06@ (1.4)

Where the two new operators ( x ^ i ( t ), p ^ i ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGdaqadaqaaiaa dshaaiaawIcacaGLPaaacaGGSaGabmiCayaajaWaaSbaaKqbGeaaca WGPbaabeaajuaGdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIca caGLPaaaaaa@435E@ in Heisenberg picture are related to the corresponding two new operators ( x ^ i , p ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGcaGGSaGabmiC ayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIcacaGLPaaaaa a@3E5A@ in Schrödinger picture from the two projections relations:

x ^ i ( t )=exp(i H nc ( t t 0 ))* x ^ i *exp(i H nc ( t t 0 ))    and    p ^ i ( t )=exp(i H nc ( t t 0 ))* p ^ i *exp(i H nc ( t t 0 )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qcamaaBaaajuaibaGaamyAaaqcfayabaWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaeyypa0JaciyzaiaacIhacaGGWbGaaiikaiaadMgaca WGibWaaSbaaKqbGeaacaWGUbGaam4yaaqabaqcfa4aaeWaaeaacaWG 0bGaeyOeI0IaamiDamaaBaaajuaibaGaaGimaaqcfayabaaacaGLOa GaayzkaaGaaiykaKqbGiaacQcajuaGceWG4bGbaKaadaWgaaqcfasa aiaadMgaaeqaaKqbakaacQcaciGGLbGaaiiEaiaacchacaGGOaGaey OeI0IaamyAaiaadIeadaWgaaqcfasaaiaad6gacaWGJbaajuaGbeaa daqadaqaaiaadshacqGHsislcaWG0bWaaSbaaKqbGeaacaaIWaaaju aGbeaaaiaawIcacaGLPaaacaGGPaGaaeiiaiaabccacaqGGaGaaeii aiaabggacaqGUbGaaeizaiaabccacaqGGaGaaeiiaiqadchagaqcam aaBaaajuaibaGaamyAaaqabaqcfa4aaeWaaeaacaWG0baacaGLOaGa ayzkaaGaeyypa0JaciyzaiaacIhacaGGWbGaaiikaiaadMgacaWGib WaaSbaaKqbGeaacaWGUbGaam4yaaqcfayabaWaaeWaaeaacaWG0bGa eyOeI0IaamiDamaaBaaajuaibaGaaGimaaqcfayabaaacaGLOaGaay zkaaGaaiykaKqbGiaacQcajuaGceWGWbGbaKaadaWgaaqcfasaaiaa dMgaaeqaaiaacQcajuaGciGGLbGaaiiEaiaacchacaGGOaGaeyOeI0 IaamyAaiaadIeadaWgaaqcfasaaiaad6gacaWGJbaajuaGbeaadaqa daqaaiaadshacqGHsislcaWG0bWaaSbaaKqbGeaacaaIWaaajuaGbe aaaiaawIcacaGLPaaacaGGPaaaaa@90AC@ (1.5)

Here H nc MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbaajuaGbeaaaaa@39FE@ denote to the new quantum Hamiltonian operator in the symmetries of (NC: 2D-RSP) and (NC: 3D-RSP). The very small two parameters θ μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXn aaCaaabeqcfasaaiabeY7aTjabe27aUbaaaaa@3BED@ and θ ¯ μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaaba GaeqiUdehaamaaCaaabeqcfasaaiabeY7aTjabe27aUbaaaaa@3BFE@ (compared to the energy) are elements of two ant symmetric real matrixes and (*) denote to the new star product, which is generalized between two arbitrary functions f( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadIhacaGGSaGaamiCaaGaayjkaiaawMcaaaaa@3B8F@ and g( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgada qadaqaaiaadIhacaGGSaGaamiCaaGaayjkaiaawMcaaaaa@3B90@ to ( fg )( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOzaiabgEHiQiaadEgaaiaawIcacaGLPaaadaqadaqaaiaadIha caGGSaGaamiCaaGaayjkaiaawMcaaaaa@3EF3@ instead of the usual product ( fg )( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOzaiaadEgaaiaawIcacaGLPaaadaqadaqaaiaadIhacaGGSaGa amiCaaGaayjkaiaawMcaaaaa@3E04@ in ordinary (two-three) dimensional spaces [39-63]:

( fg )( x,p )  exp( i 2 θ μν μ x ν x + i 2 θ ¯ μν μ p ν p )( fg )( x,p ) =  (fg i 2 θ μν μ x f ν x g i 2 θ ¯ μν μ p f ν p g)( x,p )| ( x μ = x ν , p μ = p ν ) +O( θ 2 , θ ¯ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOzaKqbGiabgEHiQKqbakaadEgaaiaawIcacaGLPaaadaqadaqa aiaadIhacaGGSaGaamiCaaGaayjkaiaawMcaaabaaaaaaaaapeGaai iOa8aacqGHHjIUpeGaaiiOa8aaciGGLbGaaiiEaiaacchacaGGOaWa aSaaaeaacaWGPbaabaGaaGOmaaaacqaH4oqCdaahaaqabKqbGeaacq aH8oqBcqaH9oGBaaqcfaOaeyOaIy7aa0baaKqbGeaacqaH8oqBaeaa caWG4baaaKqbakabgkGi2oaaDaaajuaibaGaeqyVd4gabaGaamiEaa aajuaGcqGHRaWkdaWcaaqaaiaadMgaaeaacaaIYaaaamaanaaabaGa eqiUdehaamaaCaaajuaibeqaaiabeY7aTjabe27aUbaajuaGcqGHci ITdaqhaaqcfasaaiabeY7aTbqaaiaadchaaaqcfaOaaGPaVlabgkGi 2oaaDaaajuaibaGaeqyVd4gabaGaamiCaaaajuaGcaGGPaWaaeWaae aacaWGMbqcfaIaam4zaaqcfaOaayjkaiaawMcaamaabmaabaGaamiE aiaacYcacaWGWbaacaGLOaGaayzkaaWdbiaacckapaGaeyypa0Zaaq GaaeaapeGaaiiOa8aacaGGOaGaamOzaKqbGiaadEgajuaGcqGHsisl daWcaaqaaiaadMgaaeaacaaIYaaaaiabeI7aXnaaCaaajuaibeqaai abeY7aTjabe27aUbaajuaGcqGHciITdaqhaaqcfasaaiabeY7aTbqa aiaadIhaaaqcfaOaamOzaiabgkGi2oaaDaaajuaibaGaeqyVd4gaba GaamiEaaaajuaGcaWGNbGaeyOeI0YaaSaaaeaacaWGPbaabaGaaGOm aaaadaqdaaqaaiabeI7aXbaadaahaaqabKqbGeaacqaH8oqBcqaH9o GBaaqcfaOaeyOaIy7aa0baaKqbGeaacqaH8oqBaeaacaWGWbaaaKqb akaadAgacaaMc8UaeyOaIy7aa0baaKqbGeaacqaH9oGBaeaacaWGWb aaaKqbakaadEgacaGGPaWaaeWaaeaacaWG4bGaaiilaiaadchaaiaa wIcacaGLPaaaaiaawIa7amaaBaaabaWaaeWaaeaacaWG4bWaaWbaae qajuaibaGaeqiVd0gaaKqbakabg2da9iaadIhadaahaaqcfasabeaa cqaH9oGBaaqcfaOaaiilaiaadchadaahaaqabKqbGeaacqaH8oqBaa qcfaOaeyypa0JaamiCamaaCaaabeqcfasaaiabe27aUbaaaKqbakaa wIcacaGLPaaaaeqaaiabgUcaRiaad+eadaqadaqaaiabeI7aXnaaCa aabeqcfasaaiaaikdaaaqcfaOaaiilamaanaaabaGaeqiUdehaamaa CaaajuaqbeqaaiaaikdaaaaajuaGcaGLOaGaayzkaaaaaa@CB52@ (2)

Where the two covariant derivatives ( μ x f( x,p ), μ p f( x,p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeyOaIy7aa0baaeaacqaH8oqBaeaacaWG4baaaiaadAgadaqadaqa aiaadIhacaGGSaGaamiCaaGaayjkaiaawMcaaiaacYcacqGHciITda qhaaqaaiabeY7aTbqaaiaadchaaaGaamOzamaabmaabaGaamiEaiaa cYcacaWGWbaacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@4B4C@ are denotes to the ( f( x,p ) x μ , f( x,p ) p μ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaSaaaeaacqGHciITcaWGMbWaaeWaaeaacaWG4bGaaiilaiaadcha aiaawIcacaGLPaaaaeaacqGHciITcaWG4bWaaWbaaeqajuaibaGaeq iVd0gaaaaajuaGcaGGSaWaaSaaaeaacqGHciITcaWGMbWaaeWaaeaa caWG4bGaaiilaiaadchaaiaawIcacaGLPaaaaeaacqGHciITcaWGWb WaaWbaaeqajuaibaGaeqiVd0gaaaaaaKqbakaawIcacaGLPaaaaaa@4FB0@ , respectively, and the two following terms [ i 2 θ μν μ x f( x,p ) ν x g( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTm aalaaabaGaamyAaaqaaiaaikdaaaGaeqiUde3aaWbaaeqajuaibaGa eqiVd0MaeqyVd4gaaKqbakabgkGi2oaaDaaajuaibaGaeqiVd0gaba GaamiEaaaajuaGcaWGMbWaaeWaaeaacaWG4bGaaiilaiaadchaaiaa wIcacaGLPaaacqGHciITdaqhaaqcfasaaiabe27aUbqaaiaadIhaaa qcfaOaam4zamaabmaabaGaamiEaiaacYcacaWGWbaacaGLOaGaayzk aaaaaa@533F@ , i 2 θ ¯ μν μ p f( x,p ) ν p g( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTm aalaaabaGaamyAaaqaaiaaikdaaaWaa0aaaKqbGeaacqaH4oqCaaqc fa4aaWbaaeqajuaibaGaeqiVd0MaeqyVd4gaaKqbakabgkGi2oaaDa aajuaibaGaeqiVd0gabaGaamiCaaaajuaGcaWGMbWaaeWaaeaacaWG 4bGaaiilaiaadchaaiaawIcacaGLPaaacaaMc8UaeyOaIy7aa0baaK qbGeaacqaH9oGBaeaacaWGWbaaaKqbakaadEgadaqadaqaaiaadIha caGGSaGaamiCaaGaayjkaiaawMcaaaaa@5587@ ] are induced by (space-space) and (phase-phase) noncommutativity properties, respectively, a Boopp's shift method can be used, instead of solving any quantum systems by using directly star product procedure [39-66]:

[ x ^ i , x ^ j ]=i θ ij and [ p ^ i , p ^ j ]=i θ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba GabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGcaGGSaGabmiE ayaajaWaaSbaaKqbGeaacaWGQbaajuaGbeaaaiaawUfacaGLDbaacq GH9aqpcaWGPbGaeqiUde3aaSbaaKqbGeaacaWGPbGaamOAaaqabaqc faybaeqabeGaaaqaaiaabggacaqGUbGaaeizaaqaamaadmaabaGabm iCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGcaGGSaGabmiCayaa jaWaaSbaaKqbGeaacaWGQbaajuaGbeaaaiaawUfacaGLDbaacqGH9a qpcaWGPbWaa0aaaeaacqaH4oqCaaWaaSbaaKqbGeaacaWGPbGaamOA aaqcfayabaaaaaaa@56B1@ (3.1)

The, four generalized positions and momentum coordinates in the noncommutative quantum mechanics ( x ^ , y ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmiEayaajaGaaiilaiqadMhagaqcaaGaayjkaiaawMcaaaaa@3ACD@ and ( p ^ x , p ^ y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmiCayaajaWaaSbaaKqbGeaacaWG4baabeaajuaGcaGGSaGabmiC ayaajaWaaSbaaKqbGeaacaWG5baajuaGbeaaaiaawIcacaGLPaaaaa a@3E71@ are depended with corresponding four usual generalized positions and momentum coordinates in the usual quantum mechanics ( x,y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamiEaiaacYcacaWG5baacaGLOaGaayzkaaaaaa@3AAD@ and ( p x , p y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamiCamaaBaaajuaibaGaamiEaaqabaqcfaOaaiilaiaadchadaWg aaqcfasaaiaadMhaaKqbagqaaaGaayjkaiaawMcaaaaa@3E51@ by the following four relations [32-55]:

{ x ^ =x θ 2 p y , y ^ =y+ θ 2 p x p ^ x = p x + θ ¯ 2 y  and p ^ y = p x θ ¯ 2 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaea qabeaaceWG4bGbaKaacqGH9aqpcaWG4bGaeyOeI0YaaSaaaeaacqaH 4oqCaeaacaaIYaaaaiaadchadaWgaaqcfasaaiaadMhaaKqbagqaau aabeqabiaaaeaacaGGSaaabaGabmyEayaajaGaeyypa0JaamyEaiab gUcaRmaalaaabaGaeqiUdehabaGaaGOmaaaacaWGWbWaaSbaaKqbGe aacaWG4baabeaaaaaajuaGbaGabmiCayaajaWaaSbaaKqbGeaacaWG 4baabeaajuaGcqGH9aqpcaWGWbWaaSbaaKqbGeaacaWG4baajuaGbe aacqGHRaWkdaWcaaqaamaanaaabaGaeqiUdehaaaqaaiaaikdaaaGa amyEauaabeqabiaaaeaacaqGGaGaaeyyaiaab6gacaqGKbaabaGabm iCayaajaWaaSbaaKqbGeaacaWG5baajuaGbeaacqGH9aqpcaWGWbWa aSbaaKqbGeaacaWG4baabeaajuaGcqGHsisldaWcaaqaamaanaaaba GaeqiUdehaaaqaaiaaikdaaaGaamiEaaaaaaGaay5Eaaaaaa@64B0@ (3.2)

{ x ^ =x θ 12 2 p y θ 13 2 p z , y ^ =y θ 21 2 p x θ 23 2 p z and    z ^ =z θ 31 2 p x θ 32 2 p y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaea qabeaaceWG4bGbaKaacqGH9aqpcaWG4bGaeyOeI0YaaSaaaeaacqaH 4oqCdaWgaaqcfasaaiaaigdacaaIYaaajuaGbeaaaeaacaaIYaaaai aadchadaWgaaqcfasaaiaadMhaaKqbagqaaiabgkHiTmaalaaabaGa eqiUde3aaSbaaKqbGeaacaaIXaGaaG4maaqabaaajuaGbaGaaGOmaa aacaWGWbWaaSbaaKqbGeaacaWG6baabeaajuaGcaGGSaGabmyEayaa jaGaeyypa0JaamyEaiabgkHiTmaalaaabaGaeqiUde3aaSbaaKqbGe aacaaIYaGaaGymaaqcfayabaaabaGaaGOmaaaacaWGWbWaaSbaaKqb GeaacaWG4baajuaGbeaacqGHsisldaWcaaqaaiabeI7aXnaaBaaaju aibaGaaGOmaiaaiodaaeqaaaqcfayaaiaaikdaaaGaamiCamaaBaaa juaibaGaamOEaaqcfayabaaabaGaaeyyaiaab6gacaqGKbGaaeiiai aabccacaqGGaGabmOEayaajaGaeyypa0JaamOEaiabgkHiTmaalaaa baGaeqiUde3aaSbaaKqbGeaacaaIZaGaaGymaaqcfayabaaabaGaaG OmaaaacaWGWbWaaSbaaKqbGeaacaWG4baabeaajuaGcqGHsisldaWc aaqaaiabeI7aXnaaBaaajuaibaGaaG4maiaaikdaaKqbagqaaaqaai aaikdaaaGaamiCamaaBaaajuaibaGaamyEaaqcfayabaaaaiaawUha aaaa@799D@ (3.3)

and

{ p ^ x = p x θ ¯ 12 2 y θ ¯ 13 2 z, p ^ y = p y θ ¯ 21 2 x θ ¯ 23 2 z and     p ^ z = p z θ ¯ 31 2 x θ ¯ 32 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaea qabeaaceWGWbGbaKaadaWgaaqcfasaaiaadIhaaeqaaKqbakabg2da 9iaadchadaWgaaqcfasaaiaadIhaaeqaaKqbakabgkHiTmaalaaaba Waa0aaaeaacqaH4oqCaaWaaSbaaKqbGeaacaaIXaGaaGOmaaqcfaya baaabaGaaGOmaaaacaWG5bGaeyOeI0YaaSaaaeaadaqdaaqaaiabeI 7aXbaadaWgaaqcfasaaiaaigdacaaIZaaabeaaaKqbagaacaaIYaaa aiaadQhacaGGSaGabmiCayaajaWaaSbaaKqbGeaacaWG5baajuaGbe aacqGH9aqpcaWGWbWaaSbaaKqbGeaacaWG5baajuaGbeaacqGHsisl daWcaaqaamaanaaabaGaeqiUdehaamaaBaaabaWaaSbaaKqbGeaaca aIYaGaaGymaaqcfayabaaabeaaaeaacaaIYaaaaiaadIhacqGHsisl daWcaaqaamaanaaabaGaeqiUdehaamaaBaaabaWaaSbaaKqbGeaaca aIYaGaaG4maaqcfayabaaabeaaaeaacaaIYaaaaiaadQhaaeaacaqG HbGaaeOBaiaabsgacaqGGaGaaeiiaiaabccacaqGGaGabmiCayaaja WaaSbaaKqbGeaacaWG6baajuaGbeaacqGH9aqpcaWGWbWaaSbaaKqb GeaacaWG6baabeaajuaGcqGHsisldaWcaaqaamaanaaabaGaeqiUde haamaaBaaabaWaaSbaaKqbGeaacaaIZaGaaGymaaqcfayabaaabeaa aeaacaaIYaaaaiaadIhacqGHsisldaWcaaqaamaanaaabaGaeqiUde haamaaBaaajuaibaqcfa4aaSbaaKqbGeaacaaIZaGaaGOmaaqabaaa beaaaKqbagaacaaIYaaaaiaadMhaaaGaay5Eaaaaaa@7BE6@ (3.4)

The non-vanish 9-commutators in (NC-2D: RSP) and (NC-3D: RSP) can be determined as follows:

[ x ^ , p ^ x ]=[ y ^ , p ^ y ]=i, [ x ^ , y ^ ]=i θ 12              and                    [ p ^ x , p ^ y ]=i θ ¯ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aam WaaeaaceWG4bGbaKaacaGGSaGabmiCayaajaWaaSbaaKqbGeaacaWG 4baajuaGbeaaaiaawUfacaGLDbaacqGH9aqpdaWadaqaaiqadMhaga qcaiaacYcaceWGWbGbaKaadaWgaaqcfasaaiaadMhaaKqbagqaaaGa ay5waiaaw2faaiabg2da9iaadMgacaGGSaaakeaajuaGdaWadaqaai qadIhagaqcaiaacYcaceWG5bGbaKaaaiaawUfacaGLDbaacqGH9aqp caWGPbGaeqiUde3aaSbaaKqbGeaacaaIXaGaaGOmaaqabaqcfaOaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabggacaqGUbGaaeizaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaWaamWaaeaaceWGWbGbaKaadaWgaaqcfasa aiaadIhaaKqbagqaaiaacYcaceWGWbGbaKaadaWgaaqcfasaaiaadM haaKqbagqaaaGaay5waiaaw2faaiabg2da9iaadMgadaqdaaqaaiab eI7aXbaadaWgaaqcfasaaiaaigdacaaIYaaajuaGbeaaaaaa@78CA@ (3.5)

and

[ x ^ , p ^ x ]=[ y ^ , p ^ y ]=[ z ^ , p ^ z ]=i, [ x ^ , y ^ ]=i θ 12 ,[ x ^ , z ^ ]=i θ 13 ,[ y ^ , z ^ ]=i θ 23 [ p ^ x , p ^ y ]=i θ ¯ 12 ,[ p ^ y , p ^ z ]=i θ ¯ 23 ,[ p ^ x , p ^ z ]=i θ ¯ 13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aam WaaeaaceWG4bGbaKaacaGGSaGabmiCayaajaWaaSbaaKqbGeaacaWG 4baabeaaaKqbakaawUfacaGLDbaacqGH9aqpdaWadaqaaiqadMhaga qcaiaacYcaceWGWbGbaKaadaWgaaqcfasaaiaadMhaaKqbagqaaaGa ay5waiaaw2faaiabg2da9maadmaabaGabmOEayaajaGaaiilaiqadc hagaqcamaaBaaajuaibaGaamOEaaqcfayabaaacaGLBbGaayzxaaGa eyypa0JaamyAaiaacYcaaeaadaWadaqaaiqadIhagaqcaiaacYcace WG5bGbaKaaaiaawUfacaGLDbaacqGH9aqpcaWGPbGaeqiUde3aaSba aKqbGeaacaaIXaGaaGOmaaqcfayabaGaaiilamaadmaabaGabmiEay aajaGaaiilaiqadQhagaqcaaGaay5waiaaw2faaiabg2da9iaadMga cqaH4oqCdaWgaaqcfasaaiaaigdacaaIZaaajuaGbeaacaGGSaWaam WaaeaaceWG5bGbaKaacaGGSaGabmOEayaajaaacaGLBbGaayzxaaGa eyypa0JaamyAaiabeI7aXnaaBaaajuaibaGaaGOmaiaaiodaaeqaaa Gcbaqcfa4aamWaaeaaceWGWbGbaKaadaWgaaqcfasaaiaadIhaaeqa aKqbakaacYcaceWGWbGbaKaadaWgaaqcfasaaiaadMhaaKqbagqaaa Gaay5waiaaw2faaiabg2da9iaadMgadaqdaaqaaiabeI7aXbaadaWg aaqcfasaaiaaigdacaaIYaaabeaajuaGcaGGSaWaamWaaeaaceWGWb GbaKaadaWgaaqcfasaaiaadMhaaeqaaKqbakaacYcaceWGWbGbaKaa daWgaaqcfasaaiaadQhaaeqaaaqcfaOaay5waiaaw2faaiabg2da9i aadMgadaqdaaqaaiabeI7aXbaadaWgaaqcfasaaiaaikdacaaIZaaa beaajuaGcaGGSaWaamWaaeaaceWGWbGbaKaadaWgaaqcfasaaiaadI haaKqbagqaaiaacYcaceWGWbGbaKaadaWgaaqcfasaaiaadQhaaKqb agqaaaGaay5waiaaw2faaiabg2da9iaadMgadaqdaaqaaiabeI7aXb aadaWgaaqcfasaaiaaigdacaaIZaaabeaaaaaa@9CAA@ (3.6)

Which allow us to getting the two operators r ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadkhaga qcamaaCaaabeqcfasaaiaaikdaaaaaaa@388C@ and p ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaCaaabeqcfasaaiaaikdaaaaaaa@388A@ on a noncommutative two dimensional space-phase as follows [32-48]:

r ^ 2 = r 2 θ L z     and     p ^ 2 = p 2 + θ ¯ L z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadkhaga qcamaaCaaajuaibeqaaiaaikdaaaqcfaOaeyypa0JaamOCamaaCaaa beqcfasaaiaaikdaaaqcfaOaeyOeI0IaeqiUdeNaamitamaaBaaaju aibaGaamOEaaqcfayabaGaaeiiaiaabccacaqGGaGaaeiiaiaabgga caqGUbGaaeizaiaabccacaqGGaGaaeiiaiaabccaceWGWbGbaKaada ahaaqcfasabeaacaaIYaaaaKqbakabg2da9iaadchadaahaaqcfasa beaacaaIYaaaaKqbakabgUcaRiqbeI7aXzaaraGaamitamaaBaaaju aibaGaamOEaaqcfayabaaaaa@5566@ (4.1)

r ^ 2 = r 2 L Θ        and                 p ^ 2 2μ  =  p 2 2μ  +  L θ ¯ 2μ   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadkhaga qcamaaCaaajuaibeqaaiaaikdaaaqcfaOaeyypa0JaamOCamaaCaaa juaibeqaaiaaikdaaaqcfaOaeyOeI0ccbeGab8htayaalaGafuiMde LbaSaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeyyaiaab6gacaqGKbGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiamaalaaabaGabmiCayaajaWaaWbaaeqajuaiba GaaGOmaaaaaKqbagaacaqGYaGaeqiVd0gaaiaabccacqGH9aqpcaqG GaWaaSaaaeaacaWGWbWaaWbaaKqbGeqabaGaaGOmaaaaaKqbagaaca qGYaGaeqiVd0gaaiaabccacqGHRaWkcaqGGaWaaSaaaeaaceWFmbGb aSaadaWhcaqaamaanaaabaacceGae4hUdehaaaGaay51GaaabaGaae OmaiabeY7aTbaacaqGGaaaaa@6786@ (4.2)

Where the two couplings LΘ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqbakaa=X eacqqHyoquaaa@38C7@ and L θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqbakqa=X eagaWcamaaFiaabaWaa0aaaeaaiiqacqGF4oqCaaaacaGLxdcaaaa@3AE2@ are given by, respectively:

LΘ L x Θ 12 + L y Θ 23 + L z Θ 13     and     L θ ¯ L x θ ¯ 12 + L y θ ¯ 23 + L z θ ¯ 13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqbakaa=X eacqqHyoqucqGHHjIUcaWGmbWaaSbaaKqbGeaacaWG4baajuaGbeaa cqqHyoqudaWgaaqcfasaaiaaigdacaaIYaaajuaGbeaacqGHRaWkca WGmbWaaSbaaKqbGeaacaWG5baajuaGbeaacqqHyoqudaWgaaqcfasa aiaaikdacaaIZaaajuaGbeaacqGHRaWkcaWGmbWaaSbaaKqbGeaaca WG6baabeaajuaGcqqHyoqudaWgaaqcfasaaiaaigdacaaIZaaajuaG beaacaqGGaGaaeiiaiaabccacaqGGaGaaeyyaiaab6gacaqGKbGaae iiaiaabccacaqGGaGaaeiiaiqa=XeagaWcamaaFiaabaWaa0aaaeaa iiqacqGF4oqCaaaacaGLxdcacqGHHjIUcaWGmbWaaSbaaKqbGeaaca WG4baabeaajuaGdaqdaaqaaiabeI7aXbaadaWgaaqcfasaaiaaigda caaIYaaabeaajuaGcqGHRaWkcaWGmbWaaSbaaKqbGeaacaWG5baaju aGbeaadaqdaaqaaiabeI7aXbaadaWgaaqcfasaaiaaikdacaaIZaaa beaajuaGcqGHRaWkcaWGmbWaaSbaaKqbGeaacaWG6baabeaajuaGda qdaaqaaiabeI7aXbaadaWgaaqcfasaaiaaigdacaaIZaaabeaaaaa@7346@ (5.1)

It is-well known, that, in quantum mechanics, the three components ( L x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada WgaaqcfasaaiaadIhaaKqbagqaaaaa@3924@ , L y   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada WgaaqcfasaaiaadMhaaKqbagqaaiaabccaaaa@39C8@ and L z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada WgaaqcfasaaiaadQhaaKqbagqaaaaa@3926@ ) are determined, in Cartesian coordinates:

L x =y p z z p y , L y  = zp x -xp z       and      L z =x p y y p x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada WgaaqcfasaaiaadIhaaeqaaKqbakabg2da9iaadMhacaWGWbWaaSba aKqbGeaacaWG6baabeaajuaGcqGHsislcaWG6bGaamiCamaaBaaaju aibaGaamyEaaqcfayabaGaaiilaiaadYeadaWgaaqcfasaaiaadMha aeqaaKqbakaabccacqGH9aqpcaqG6bGaaeiCamaaBaaajuaibaGaae iEaaqcfayabaGaaeylaiaabIhacaqGWbWaaSbaaKqbGeaacaqG6baa beaajuaGcaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGHb GaaeOBaiaabsgacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaadYea daWgaaqcfasaaiaadQhaaeqaaKqbakabg2da9iaadIhacaWGWbWaaS baaKqbGeaacaWG5baabeaajuaGcqGHsislcaWG5bGaamiCamaaBaaa juaibaGaamiEaaqcfayabaaaaa@65F1@ (5.2)

The study of inverse-square potential has now become a very interest field due to their applications in different fields [1], this work is aimed at obtaining an analytic expression for the eigenenergies of a inverse-square potential in (NC: 2D-RSP) and (NC: 3D-RSP) using the generalization Boopp’s shift method based on mentioned formalisms on above equations to discover the new symmetries and a possibility to obtain another applications to this potential in different fields, it is important to notice that, this potential was studied, in ordinary two dimensional spaces, by authors Shi-Hai Dong and Guo-Hua Sun of the Ref. the Schrödinger equation with a Coulomb plus inverse-square potential in D dimensions [1]. The organization scheme of the study is given as follows: In next section, we briefly review the Schrödinger equation with inverse-square potential on based to Ref. [1]. The Section 3, devoted to studying the (two-three) deformed Schrödinger equation by applying both Boopp's shift method to the inverse-square potential. In the fourth section and by applying standard perturbation theory we find the quantum spectrum of the excited states in (NC-2D: RSP) and (NC-3D: RSP) for spin-orbital interaction. In the next section, we derive the magnetic spectrum for studied potential. In the sixth section, we resume the global spectrum and corresponding noncommutative Hamiltonian for inverse-square potential. Finally, the important results and the conclusions are discussed in last section.

Review The Eignenfunctions and the Energy Eigenvalues for Inverse-Square Potential in Ordinary Two Dimensional Space

Here we will firstly describe the essential steps, which gives the solutions of time independent Schrödinger equation for a fermionic particle like electron of rest mass and its energy moving in inverse-square potential [1]:

V( r )= A r 2 B r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada qadaqaaiaadkhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadgea aeaacaWGYbWaaWbaaeqajuaibaGaaGOmaaaaaaqcfaOaeyOeI0YaaS aaaeaacaWGcbaabaGaamOCaaaaaaa@40FC@ (6)

Where A and B are two positive constant coefficients. The above potential is the sum of Colombian ( B r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeyOeI0YaaSaaaeaacaWGcbaabaGaamOCaaaaaiaawIcacaGLPaaa aaa@3ABD@ and inverse-square potential ( A r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaSaaaeaacaWGbbaabaGaamOCamaaCaaajuaibeqaaiaaikdaaaaa aaqcfaOaayjkaiaawMcaaaaa@3B69@ , if we insert this potential into the non-relativistic Schrödinger equation; we obtain the following equation, in two and three dimensional spaces, respectively, as follows:

{ 2 2 m 0 [ 1 r r ( r r )+ 1 r 2 2 ϕ 2 ] A r 2 + B r }Ψ( r,ϕ )= Ε 2d Ψ( r,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaacmaaba GaeyOeI0YaaSaaaeaadaqfGaqabeqajuaibaGaaGOmaaqcfayaaiab l+qiObaaaeaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfayaba aaamaadmaabaWaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiab gkGi2cqaaiabgkGi2kaadkhaaaWaaeWaaeaacaWGYbWaaSaaaeaacq GHciITaeaacqGHciITcaWGYbaaaaGaayjkaiaawMcaaiabgUcaRmaa laaabaGaaGymaaqaaiaadkhadaahaaqabKqbGeaacaaIYaaaaaaaju aGdaWcaaqaamaavacabeqabKqbGeaacaaIYaaajuaGbaGaeyOaIyla aaqaaiabgkGi2oaavacabeqabKqbGeaacaaIYaaajuaGbaGaeqy1dy gaaaaaaiaawUfacaGLDbaacqGHsisldaWcaaqaaiaadgeaaeaacaWG YbWaaWbaaeqajuaibaGaaGOmaaaaaaqcfaOaey4kaSYaaSaaaeaaca WGcbaabaGaamOCaaaaaiaawUhacaGL9baacqqHOoqwdaqadaqaaiaa dkhacaGGSaGaeqy1dygacaGLOaGaayzkaaGaeyypa0JaeuyLdu0aaS baaKqbGeaacaaIYaGaamizaaqcfayabaGaeuiQdK1aaeWaaeaacaWG YbGaaiilaiabew9aMbGaayjkaiaawMcaaaaa@72F5@ (7.1)

{ 2 2 m 0 [ 1 r 2 r ( r 2 r )+ 1 r 2 sinθ θ ( sinθ θ )+ 1 r 2 ( sinθ ) 2 2 ϕ 2 ] A r 2 + B r }Ψ( r,θ,ϕ )= Ε 3d Ψ( r,θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaacmaaba GaeyOeI0YaaSaaaeaadaqfGaqabeqajuaibaGaaGOmaaqcfayaaiab l+qiObaaaeaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfayaba aaamaadmaabaWaaSaaaeaacaaIXaaabaWaaubiaeqabeqcfasaaiaa ikdaaKqbagaacaWGYbaaaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2k aadkhaaaWaaeWaaeaadaqfGaqabeqajuaibaGaaGOmaaqcfayaaiaa dkhaaaWaaSaaaeaacqGHciITaeaacqGHciITcaWGYbaaaaGaayjkai aawMcaaiabgUcaRmaalaaabaGaaGymaaqaamaavacabeqabKqbGeaa caaIYaaajuaGbaGaamOCaaaaciGGZbGaaiyAaiaac6gacqaH4oqCaa WaaSaaaeaacqGHciITaeaacqGHciITcqaH4oqCaaWaaeWaaeaaciGG ZbGaaiyAaiaac6gacqaH4oqCdaWcaaqaaiabgkGi2cqaaiabgkGi2k abeI7aXbaaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaigdaaeaa daqfGaqabeqajuaibaGaaGOmaaqcfayaaiaadkhaaaWaaubiaeqabe qcfasaaiaaikdaaKqbagaadaqadaqaaiGacohacaGGPbGaaiOBaiab eI7aXbGaayjkaiaawMcaaaaaaaWaaSaaaeaadaqfGaqabeqajuaiba GaaGOmaaqcfayaaiabgkGi2caaaeaacqGHciITdaqfGaqabeqajuai baGaaGOmaaqcfayaaiabew9aMbaaaaaacaGLBbGaayzxaaGaeyOeI0 YaaSaaaeaacaWGbbaabaGaamOCamaaCaaabeqcfasaaiaaikdaaaaa aKqbakabgUcaRmaalaaabaGaamOqaaqaaiaadkhaaaaacaGL7bGaay zFaaGaeuiQdK1aaeWaaeaacaWGYbGaaiilaiabeI7aXjaacYcacqaH vpGzaiaawIcacaGLPaaacqGH9aqpcqqHvoqrdaWgaaqcfasaaiaaio dacaWGKbaabeaajuaGcqqHOoqwdaqadaqaaiaadkhacaGGSaGaeqiU deNaaiilaiabew9aMbGaayjkaiaawMcaaaaa@9B3A@ (7.2)

Here Ψ( r,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaamOCaiaacYcacqaHvpGzaiaawIcacaGLPaaaaaa@3D00@ and Ψ( r,θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaamOCaiaacYcacqaH4oqCcaGGSaGaeqy1dygacaGLOaGa ayzkaaaaaa@3F66@ is the solution in the (2-3) dimensional in (polar and spherical) coordinates, the complete wave function ( Ψ( r,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaamOCaiaacYcacqaHvpGzaiaawIcacaGLPaaaaaa@3D00@ and Ψ( r,θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaamOCaiaacYcacqaH4oqCcaGGSaGaeqy1dygacaGLOaGa ayzkaaaaaa@3F66@ separated as follows:

Ψ( r,ϕ )= R l ( r ) e ±iϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaamOCaiaacYcacqaHvpGzaiaawIcacaGLPaaacqGH9aqp daqfqaqabeaacaWGSbaabeqaaiaadkfaaaWaaeWaaeaacaWGYbaaca GLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgglaXkaadMgacqaH vpGzaaaaaa@485A@ (8.1)

and

Ψ( x )= R l ( r ) Y l l ( θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maavababeqcfasa aiaadYgaaKqbagqabaGaamOuaaaadaqadaqaaiaadkhaaiaawIcaca GLPaaadaqfWaqabKqbGeaacaWGSbaabaGaamiBaaqcfayaaiaadMfa aaWaaeWaaeaacqaH4oqCcaGGSaGaeqy1dygacaGLOaGaayzkaaaaaa@4A46@ (8.2)

Substituting eq. (8.1) and (8.2) into eq. (7.1) and (7.2), we obtain the radial function R l ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaWGSbaabeGcbaGaamOuaaaadaqadaqaaiaadkhaaiaawIcacaGL Paaaaaa@3A81@ satisfied the following equation, in (two-three) dimensional spaces [1]:

d 2 R l ( ρ ) d ρ 2 + 2 ρ dR l ( ρ ) dρ +( 1 4 + τ ρ 2A+ l 2 ρ 2 ) R l ( r ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaubeaeqajuaibaGaamiBaaqcfayabeaacaWGKbWaaWbaaKqbGeqa baGaaGOmaaaajuaGcaWGsbaaamaabmaabaGaeqyWdihacaGLOaGaay zkaaaabaGaamizaiabeg8aYnaaCaaajuaibeqcKvaq=haacaaIYaaa aaaajuaGcqGHRaWkdaWcaaqaaiaaikdaaeaacqaHbpGCaaWaaSaaae aadaqfqaqabKqbGeaacaWGSbaajuaGbeqaaiaadsgacaWGsbaaamaa bmaabaGaeqyWdihacaGLOaGaayzkaaaabaGaamizaiabeg8aYbaacq GHRaWkdaqadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaGa ey4kaSYaaSaaaeaacqaHepaDaeaacqaHbpGCaaGaeyOeI0YaaSaaae aacaaIYaGaamyqaiabgUcaRiaadYgadaahaaqcfasabeaacaaIYaaa aaqcfayaaiabeg8aYnaaCaaajuaibeqaaiaaikdaaaaaaaqcfaOaay jkaiaawMcaamaavababeqcfasaaiaadYgaaKqbagqabaGaamOuaaaa daqadaqaaiaadkhaaiaawIcacaGLPaaaqaaaaaaaaaWdbiaacckapa Gaeyypa0ZdbiaacckapaGaaGimaaaa@6E2C@ (9.1)

1 r 2 r ( r 2 r ) R l ( r )+[ 2( ΕV( r ) ) l( l+1 ) r 2 ] R l ( r ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaamaavacabeqabKqbGeaacaaIYaaajuaGbaGaamOCaaaa aaWaaSaaaeaacqGHciITaeaacqGHciITcaWGYbaaamaabmaabaWaau biaeqabeqcfasaaiaaikdaaKqbagaacaWGYbaaamaalaaabaGaeyOa IylabaGaeyOaIyRaamOCaaaaaiaawIcacaGLPaaadaqfqaqabKqbGe aacaWGSbaajuaGbeqaaiaadkfaaaWaaeWaaeaacaWGYbaacaGLOaGa ayzkaaGaey4kaSYaamWaaeaacaaIYaWaaeWaaeaacqqHvoqrcqGHsi slcaWGwbWaaeWaaeaacaWGYbaacaGLOaGaayzkaaaacaGLOaGaayzk aaGaeyOeI0YaaSaaaeaacaWGSbWaaeWaaeaacaWGSbGaey4kaSIaaG ymaaGaayjkaiaawMcaaaqaamaavacabeqabKqbGeaacaaIYaaajuaG baGaamOCaaaaaaaacaGLBbGaayzxaaWaaubeaeqajuaibaGaamiBaa qcfayabeaacaWGsbaaamaabmaabaGaamOCaaGaayjkaiaawMcaaaba aaaaaaaapeGaaiiOa8aacqGH9aqppeGaaiiOa8aacaaIWaaaaa@6802@ (9.2)

Here ρ=r 8E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYj abg2da9iaadkhadaGcaaqaaiabgkHiTiaaiIdacaWGfbaabeaaaaa@3CBF@

and τ=B 1 2E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaey ypa0JaamOqamaakaaabaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOm aiaadweaaaaaleqaaaaa@3CE1@ . The proposed solutions of eqs. (9.1) and (9.2) are determined from the unifed relation:

R l ( ρ )= ρ λ e ρ 2 F( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaWGSbaabeGcbaGaamOuaaaadaqadaqaaiabeg8aYbGaayjkaiaa wMcaaiabg2da9maavacabeWcbeqaaiabeU7aSbGcbaGaeqyWdihaam aavacabeWcbeqaaiabgkHiTmaalaaabaGaeqyWdihabaGaaGOmaaaa aOqaaiaadwgaaaGaamOramaabmaabaGaeqyWdihacaGLOaGaayzkaa aaaa@48E1@ (10)

where λ= 2D+ 2A+ k 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSj abg2da9maalaaabaGaaGOmaiabgkHiTiaadseacqGHRaWkdaGcaaqa aiaaikdacaWGbbGaey4kaSIaam4AamaaCaaajuaibeqaaiaaikdaaa aajuaGbeaaaeaacaaIYaaaaaaa@4251@ and k=2l+D2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacq GH9aqpcaaIYaGaamiBaiabgUcaRiaadseacqGHsislcaaIYaaaaa@3D70@ . We Companie between eqs. (9.1), (9.2) and (10) to obtains [1]:

ρ d 2 F( ρ ) d ρ 2 +( 2λ+D1ρ )+ dF( ρ ) dρ +( τλ D1 2 )F( ρ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aalaaabaWaaubiaeqabeqcfasaaiaaikdaaKqbagaacaWGKbaaaiaa dAeadaqadaqaaiabeg8aYbGaayjkaiaawMcaaaqaaiaadsgadaqfGa qabeqajuaibaGaaGOmaaqcfayaaiabeg8aYbaaaaGaey4kaSYaaeWa aeaacaaIYaGaeq4UdWMaey4kaSIaamiraiabgkHiTiaaigdacqGHsi slcqaHbpGCaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaadsgacaWG gbWaaeWaaeaacqaHbpGCaiaawIcacaGLPaaaaeaacaWGKbGaeqyWdi haaiabgUcaRmaabmaabaGaeqiXdqNaeyOeI0Iaeq4UdWMaeyOeI0Ya aSaaaeaacaWGebGaeyOeI0IaaGymaaqaaiaaikdaaaaacaGLOaGaay zkaaGaamOramaabmaabaGaeqyWdihacaGLOaGaayzkaaGaeyypa0Ja aGimaaaa@679D@ (11)

The confluent hypergeometric functions φ( λτ+1/2,2λ+1;ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQn aabmaabaGaeq4UdWMaeyOeI0IaeqiXdqNaey4kaSIaaGymaiaac+ca caaIYaGaaiilaiaaikdacqaH7oaBcqGHRaWkcaaIXaGaai4oaiabeg 8aYbGaayjkaiaawMcaaaaa@486D@ are present the solutions of eq. (11) [1] :

R( ρ )=Ν ρ λ e ρ 2 φ( λτ+(D1)/2,2λ+D1;ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfada qadaqaaiabeg8aYbGaayjkaiaawMcaaiabg2da9iabf25aonaavaca beqabKqbGeaacqaH7oaBaKqbagaacqaHbpGCaaWaaubiaeqabeqaai abgkHiTmaalaaabaGaeqyWdihabaGaaGOmaaaaaeaacaWGLbaaaiab eA8aQnaabmaabaGaeq4UdWMaeyOeI0IaeqiXdqNaey4kaSIaaiikai aadseacqGHsislcaaIXaGaaiykaiaac+cacaaIYaGaaiilaiaaikda cqaH7oaBcqGHRaWkcaWGebGaeyOeI0IaaGymaiaacUdacqaHbpGCai aawIcacaGLPaaaaaa@5CE0@ (12)

The constraint conditions on the potential parameters are determined from relations [1]:

τλ(D1)/2= n =0,1,2,....... n= n + κ 2 D/2+2= n +l+1 τ=Β 1 2Ε =nl1+λ+(D1)/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaeq iXdqNaeyOeI0Iaeq4UdWMaeyOeI0IaaiikaiaadseacqGHsislcaaI XaGaaiykaiaac+cacaaIYaGaeyypa0JabmOBayaafaGaeyypa0JaaG imaiaacYcacaaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6cacaGG UaGaaiOlaiaac6cacaGGUaGaaiOlaaqaaiaad6gacqGH9aqpceWGUb GbauaacqGHRaWkdaWcaaqaaiabeQ7aRbqaaiaaikdaaaGaeyOeI0Ia amiraiaac+cacaaIYaGaey4kaSIaaGOmaiabg2da9iqad6gagaqbai abgUcaRiaadYgacqGHRaWkcaaIXaaakeaajuaGcqaHepaDcqGH9aqp cqqHsoGqdaGcaaqaamaalaaabaGaaGymaaqaaiabgkHiTiaaikdacq qHvoqraaaabeaacqGH9aqpcaWGUbGaeyOeI0IaamiBaiabgkHiTiaa igdacqGHRaWkcqaH7oaBcqGHRaWkcaGGOaGaamiraiabgkHiTiaaig dacaGGPaGaai4laiaaikdaaaaa@74A8@ (13)

The normalized wave functions Ψ( ρ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaeqyWdiNaaiilaiabew9aMbGaayjkaiaawMcaaaaa@3DC9@ expressed in terms of the radial functions and spherical harmonic functions read as [1]:

Ψ( ρ,ϕ )=( 4Β 2n2m+2 s 2 1 ) ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! ) 1/2 ρ s 2 e ρ 2 L nm1 2 s 2 ( ρ )exp( ±imϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaeqyWdiNaaiilaiabew9aMbGaayjkaiaawMcaaiabg2da 9maabmaabaWaaSaaaeaacaaI0aGaeuOKdieabaGaaGOmaiaad6gacq GHsislcaaIYaGaamyBaiabgUcaRiaaikdacaWGZbWaaSbaaeaacaaI YaaabeaacqGHsislcaaIXaaaaaGaayjkaiaawMcaamaabmaabaWaaS aaaeaadaqadaqaaiaad6gacqGHsislcaWGTbGaeyOeI0IaaGymaaGa ayjkaiaawMcaaiaacgcaaeaadaqadaqaaiaaikdacaWGUbGaeyOeI0 IaaGOmaiaad2gacqGHRaWkcaaIYaGaam4CamaaBaaajuaibaGaaGOm aaqcfayabaGaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaabaGaam OBaiabgkHiTiaad2gacqGHRaWkcaaIYaGaam4CamaaBaaajuaibaGa aGOmaaqcfayabaGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaacgcaaa aacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGymaiaac+cacaaIYaaa aKqbaoaavacabeqabeaacaWGZbWaaSbaaKqbGeaacaaIYaaabeaaaK qbagaacqaHbpGCaaWaaubiaeqabeqaamaalaaabaGaeyOeI0IaeqyW dihabaGaaGOmaaaaaeaacaWGLbaaamaavadabeqaaiaad6gacqGHsi slcaWGTbGaeyOeI0IaaGymaaqaaiaaikdacaWGZbWaaSbaaKqbGeaa caaIYaaajuaGbeaaaeaacaWGmbaaamaabmaabaGaeqyWdihacaGLOa GaayzkaaGaciyzaiaacIhacaGGWbWaaeWaaeaacqGHXcqScaWGPbGa amyBaiabew9aMbGaayjkaiaawMcaaaaa@8D0F@ (14.1)

Ψ( ρ )= ( 2B n ) 3 2 [ ( nl1 )! 2n( n+l )! ] 1 2 ρ l e ρ 2 L nl1 2l+1 ( ρ ) Y l l ( θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaGaeqyWdihacaGLOaGaayzkaaGaeyypa0Zaaubiaeqabeqc fasaaKqbaoaalaaajuaibaGaaG4maaqaaiaaikdaaaaajuaGbaWaae WaaeaadaWcaaqaaiaaikdacaWGcbaabaGaamOBaaaaaiaawIcacaGL PaaaaaWaaubiaeqabeqcfasaaKqbaoaalaaajuaibaGaaGymaaqaai aaikdaaaaajuaGbaWaamWaaeaadaWcaaqaamaabmaabaGaamOBaiab gkHiTiaadYgacqGHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaqaai aaikdacaWGUbWaaeWaaeaacaWGUbGaey4kaSIaamiBaaGaayjkaiaa wMcaaiaacgcaaaaacaGLBbGaayzxaaaaamaavacabeqabKqbGeaaca WGSbaajuaGbaGaeqyWdihaamaavacabeqabeaacqGHsisldaWcaaqa aiabeg8aYbqaaiaaikdaaaaabaGaamyzaaaadaqfWaqabKqbGeaaca WGUbGaeyOeI0IaamiBaiabgkHiTiaaigdaaeaacaaIYaGaamiBaiab gUcaRiaaigdaaKqbagaacaWGmbaaamaabmaabaGaeqyWdihacaGLOa GaayzkaaWaaubmaeqajuaibaGaamiBaaqaaiaadYgaaKqbagaacaWG zbaaamaabmaabaGaeqiUdeNaaiilaiabew9aMbGaayjkaiaawMcaaa aa@74E0@ (14.2)

And the corresponding eigenvalues Ε( n,l,D ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfw5afn aabmaabaGaamOBaiaacYcacaWGSbGaaiilaiaadseaaiaawIcacaGL Paaaaaa@3D77@ is determined from relation [1]:

Ε( n,l,D )= 2Β ( 2n2l1+ 8Α+ κ 2 ) ={ 2Β 2 ( 2n2m1+ 2A+ m 2 ) 2                                                           for D=2 2 B 2 { ( 2n ) 2 8A κ ( 2n ) 3 + 16 A 2 κ 3 ( 2n ) 3 + 48 A 2 κ 2 ( 2n ) 4 ... }  for D=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfw5afn aabmaabaGaamOBaiaacYcacaWGSbGaaiilaiaadseaaiaawIcacaGL PaaacqGH9aqpcqGHsisldaWcaaqaaiaaikdacqqHsoGqaeaadaqada qaaiaaikdacaWGUbGaeyOeI0IaaGOmaiaadYgacqGHsislcaaIXaGa ey4kaSYaaOaaaeaacaaI4aGaeuyKdeKaey4kaSYaaubiaeqabeqcfa saaiaaikdaaKqbagaacqaH6oWAaaaabeaaaiaawIcacaGLPaaaaaGa eyypa0ZaaiqaaqaabeqaaiabgkHiTmaalaaabaWaaubiaeqabeqcfa saaiaaikdaaKqbagaacaaIYaGaeuOKdieaaaqaamaavacabeqabKqb GeaacaaIYaaajuaGbaWaaeWaaeaacaaIYaGaamOBaiabgkHiTiaaik dacaWGTbGaeyOeI0IaaGymaiabgUcaRmaakaaabaGaaGOmaiaadgea cqGHRaWkcaWGTbWaaWbaaeqajuaibaGaaGOmaaaaaKqbagqaaaGaay jkaiaawMcaaaaaaaqbaeqabeGaaaqaaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGMbGaae4Bai aabkhaaeaacaWGebGaeyypa0JaaGOmaaaaaeaacqGHsislcaaIYaWa aubiaeqabeqcfasaaiaaikdaaKqbagaacaWGcbaaamaacmaabaWaau biaeqabeqcfasaaiabgkHiTiaaikdaaKqbagaadaqadaqaaiaaikda caWGUbaacaGLOaGaayzkaaaaaiabgkHiTmaalaaabaGaaGioaiaadg eaaeaacqaH6oWAaaWaaubiaeqabeqcfasaaiabgkHiTiaaiodaaKqb agaadaqadaqaaiaaikdacaWGUbaacaGLOaGaayzkaaaaaiabgUcaRm aalaaabaGaaGymaiaaiAdadaqfGaqcfasabeqabaGaaGOmaaqcfaya aiaadgeaaaaabaWaaubiaeqabeqcfasaaiaaiodaaKqbagaacqaH6o WAaaaaamaavacabeqabKqbGeaacqGHsislcaaIZaaajuaGbaWaaeWa aeaacaaIYaGaamOBaaGaayjkaiaawMcaaaaacqGHRaWkdaWcaaqaai aaisdacaaI4aWaaubiaeqabeqcfasaaiaaikdaaKqbagaacaWGbbaa aaqaamaavacabeqabKqbGeaacaaIYaaajuaGbaGaeqOUdSgaaaaada qfGaqabeqabaGaeyOeI0scfaIaaGinaaqcfayaamaabmaabaGaaGOm aiaad6gaaiaawIcacaGLPaaaaaGaeyOeI0IaaiOlaiaac6cacaGGUa aacaGL7bGaayzFaaqbaeqabeGaaaqaaiaabccacaqGMbGaae4Baiaa bkhaaeaacaWGebGaeyypa0JaaG4maaaaaaGaay5Eaaaaaa@CC86@ (15)

The rest of this section is devoted to the reapply of some essential properties of generalized Laguerre polynomials L n ( β ) ( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavadabe qcfasaaiaad6gaaeaajuaGdaqadaqcfasaaiabek7aIbGaayjkaiaa wMcaaaqcfayaaiaadYeaaaWaaeWaaeaacqaHbpGCaiaawIcacaGLPa aaaaa@4075@ which are given by:

L n ( β ) ( ρ ) 1 2i exp( ρt 1t ) ( 1t ) β+1 t n+1 dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavadabe qcfasaaiaad6gaaeaajuaGdaqadaqcfasaaiabek7aIbGaayjkaiaa wMcaaaqcfayaaiaadYeaaaWaaeWaaeaacqaHbpGCaiaawIcacaGLPa aacqGHHjIUdaWcaaqaaiaaigdaaeaacaaIYaGaey4dIuTaamyAaaaa daWdfaqaamaalaaabaGaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsi sldaWcaaqaaiabeg8aYjaadshaaeaacaaIXaGaeyOeI0IaamiDaaaa aiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGHsislcaWG0baaca GLOaGaayzkaaWaaWbaaeqajuaibaGaeqOSdiMaey4kaSIaaGymaaaa juaGcaWG0bWaaWbaaeqajuaibaGaamOBaiabgUcaRiaaigdaaaaaaK qbakaadsgacaWG0baabeqabiablgH7rlabgUIiYdaaaa@63E7@ (16)

Where β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ is integer, this can be taking the exciplicitly mathematically forms [1,65,66,67]:

L n ( β ) ( ρ )= β( β+n+1 ) n!β( β+1 ) F 1 1 ( n,β+1;ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavadabe qcfasaaiaad6gaaeaajuaGdaqadaqcfasaaiabek7aIbGaayjkaiaa wMcaaaqcfayaaiaadYeaaaWaaeWaaeaacqaHbpGCaiaawIcacaGLPa aacqGH9aqpdaWcaaqaaiabek7aInaabmaabaGaeqOSdiMaey4kaSIa amOBaiabgUcaRiaaigdaaiaawIcacaGLPaaaaeaacaWGUbGaaiyiai abek7aInaabmaabaGaeqOSdiMaey4kaSIaaGymaaGaayjkaiaawMca aaaadaWgbaqcfasaaiaaigdaaeqaaKqbaoaavababeqcfasaaiaaig daaKqbagqabaGaamOraaaadaqadaqaaiabgkHiTiaad6gacaGGSaGa eqOSdiMaey4kaSIaaGymaiaacUdacqaHbpGCaiaawIcacaGLPaaaaa a@5FA7@ (17)

The Laguerre polynomials may be defined in terms of hypergeometric functions F 1 1 ( n,β+1;ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSraaSqaai aaigdaaeqaaOWaaubeaeqaleaacaaIXaaabeGcbaGaamOraaaadaqa daqaaiabgkHiTiaad6gacaGGSaGaeqOSdiMaey4kaSIaaGymaiaacU dacqaHbpGCaiaawIcacaGLPaaaaaa@4287@ , specifically the confluent hyper geometric functions, as:

F 1 1 ( n,β+1;ρ )= n=0 a ( n ) ρ n b ( n ) n! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaBeaaba GaaGymaaqabaWaaubeaeqajuaibaGaaGymaaqcfayabeaacaWGgbaa amaabmaabaGaeyOeI0IaamOBaiaacYcacqaHYoGycqGHRaWkcaaIXa Gaai4oaiabeg8aYbGaayjkaiaawMcaaiabg2da9maaqahabaWaaSaa aeaacaWGHbWaaWbaaKqbGeqabaqcfa4aaeWaaKqbGeaacaWGUbaaca GLOaGaayzkaaaaaKqbakabeg8aYnaaCaaabeqcfasaaiaad6gaaaaa juaGbaGaamOyamaaCaaajuaibeqaaKqbaoaabmaajuaibaGaamOBaa GaayjkaiaawMcaaaaacaWGUbGaaiyiaaaaaeaacaWGUbGaeyypa0Ja aGimaaqaaiabg6HiLcqcfaOaeyyeIuoaaaa@5A22@ (18.1)

Where a ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada ahaaqabKqbGeaajuaGdaqadaqcfasaaiaad6gaaiaawIcacaGLPaaa aaaaaa@3AE7@ is the Pochhammer symbol, which can be takes the particulars values a ( 0 ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada ahaaqcfasabeaajuaGdaqadaqcfasaaiaaicdaaiaawIcacaGLPaaa aaqcfaOaeyypa0JaaGimaaaa@3CFC@ and a ( n ) =a( a+1 ).....( a+n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada ahaaqcfasabeaajuaGdaqadaqcfasaaiaad6gaaiaawIcacaGLPaaa aaqcfaOaeyypa0JaamyyamaabmaabaGaamyyaiabgUcaRiaaigdaai aawIcacaGLPaaacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlamaabmaa baGaamyyaiabgUcaRiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaa aaaa@4AD3@ , it is important to notice that, the hypergeometric functions have another common notation Φ( a,b,ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agn aabmaabaGaamyyaiaacYcacaWGIbGaaiilaiabeg8aYbGaayjkaiaa wMcaaaaa@3E69@ which considered as a function of a, b=0,1,2,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkgacq GH9aqpcaaIWaGaaiilaiabgkHiTiaaigdacaGGSaGaeyOeI0IaaGOm aiaacYcacaGGUaGaaiOlaiaac6caaaa@4097@ , and the variable . The generalized Laguerre polynomial can also be defined by the following equation:

L n ( β ) ( ρ )= i=0 n ( 1 ) i ( n+β ni ) ( i )! ρ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavadabe qcfasaaiaad6gaaeaajuaGdaqadaqcfasaaiabek7aIbGaayjkaiaa wMcaaaqcfayaaiaadYeaaaWaaeWaaeaacqaHbpGCaiaawIcacaGLPa aacqGH9aqpdaaeWbqaamaalaaabaWaaeWaaeaacqGHsislcaaIXaaa caGLOaGaayzkaaWaaWbaaKqbGeqabaGaamyAaaaajuaGdaqadaabae qabaGaamOBaiabgUcaRiabek7aIbqaaiaad6gacqGHsislcaWGPbaa aiaawIcacaGLPaaaaeaadaqadaqaaiaabMgaaiaawIcacaGLPaaaca qGHaaaaaqcfasaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqcfaOa eyyeIuoacqaHbpGCdaahaaqabKqbGeaacaWGPbaaaaaa@5AFA@ (18.2)

Deformed Schrödinger Equations and Modified Inverse-Square (m.i.s.) Potential in both (Nc-2d: rsp) and (Nc-3d: rsp):

This section is devoted to constructing of non relativistic modified Schrödinger equations (m.s.e) in both (NC-2D: RSP) and (NC-3D: RSP) for (m.i.s.) potential; to achieve this subject, we apply the essentials following steps [32-48]:

  1. Ordinary two dimensional Hamiltonian operators ( H ^ is2 ( p i , x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaabaGaamyAaiaadohajuaicaaIYaaajuaGbeaadaqadaqa aiaadchadaWgaaqcfasaaiaadMgaaeqaaKqbakaacYcacaWG4bWaaS baaKqbGeaacaWGPbaabeaaaKqbakaawIcacaGLPaaaaaa@4296@ , H ^ is3 ( p i , x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyAaiaadohacaaIZaaabeaajuaGdaqadaqa aiaadchadaWgaaqcfasaaiaadMgaaeqaaKqbakaacYcacaWG4bWaaS baaKqbGeaacaWGPbaabeaaaKqbakaawIcacaGLPaaaaaa@4297@ ) will be replaced by new two dimensional Hamiltonian operators ( H ^ nc2is ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIYaGaeyOeI0IaamyAaiaa dohaaeqaaKqbaoaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPb aabeaajuaGcaGGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaa aKqbakaawIcacaGLPaaaaaa@457E@ , H ^ nc3is ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIZaGaeyOeI0IaamyAaiaa dohaaKqbagqaamaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPb aajuaGbeaacaGGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaa aKqbakaawIcacaGLPaaaaaa@457F@ ),
  2. Ordinary complex wave function Ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaWaa8HaaeaacaWGYbaacaGLxdcaaiaawIcacaGLPaaaaaa@3C3C@ will be replacing by new complex wave function Ψ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbfI6azz aataWaaeWaaeaadaWhdaqaaiqadkhagaWeaaGaayz4GaaacaGLOaGa ayzkaaaaaa@3C7B@ ,
  3. Ordinary energies E( n,l,2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada qadaqaaiaad6gacaGGSaGaamiBaiaacYcacaaIYaaacaGLOaGaayzk aaaaaa@3CCC@ and E( n,l,3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada qadaqaaiaad6gacaGGSaGaamiBaiaacYcacaaIZaaacaGLOaGaayzk aaaaaa@3CCD@ will be replaced by new values E nc2is ( n,l,2,... ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaaGOmaiabgkHiTiaadMgacaWGZbaa juaGbeaadaqadaqaaiaad6gacaGGSaGaamiBaiaacYcacaaIYaGaai ilaiaac6cacaGGUaGaaiOlaaGaayjkaiaawMcaaaaa@45D9@ and E nc3is ( n,l,3,... ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaaG4maiabgkHiTiaadMgacaWGZbaa beaajuaGdaqadaqaaiaad6gacaGGSaGaamiBaiaacYcacaaIZaGaai ilaiaac6cacaGGUaGaaiOlaaGaayjkaiaawMcaaaaa@45DB@ , respectively.

And the last step corresponds to replace the ordinary old product by new star product , which allow us to constructing the modified two dimensional Schrödinger equation in both (NC-2D: RSP) and (NC-3D: RSP) as for (m.i.s.) potential:

H ^ nc2is ( p ^ i , x ^ i ) Ψ ( r )= E nc2is ( n,l,2,... ) Ψ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIYaGaeyOeI0IaamyAaiaa dohaaeqaaKqbaoaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPb aabeaajuaGcaGGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaa aKqbakaawIcacaGLPaaacqGHxiIkcuqHOoqwgaWeamaabmaabaWaa8 XaaeaaceWGYbGbambaaiaawgoiaaGaayjkaiaawMcaaiabg2da9iaa dweadaWgaaqaaKqbGiaad6gacaWGJbGaaGOmaiabgkHiTiaadMgaca WGZbaajuaGbeaadaqadaqaaiaad6gacaGGSaGaamiBaiaacYcacaaI YaGaaiilaiaac6cacaGGUaGaaiOlaaGaayjkaiaawMcaaiqbfI6azz aataWaaeWaaeaadaWhdaqaaiqadkhagaWeaaGaayz4GaaacaGLOaGa ayzkaaGaaGPaVdaa@6462@ (19.1)

and

H ^ nc3is ( p ^ i , x ^ i ) Ψ ( r )= E nc3is ( n,l,3,... ) Ψ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIZaGaeyOeI0IaamyAaiaa dohaaeqaaKqbaoaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPb aabeaajuaGcaGGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaajuaG beaaaiaawIcacaGLPaaacqGHxiIkcuqHOoqwgaWeamaabmaabaWaa8 XaaeaaceWGYbGbambaaiaawgoiaaGaayjkaiaawMcaaiabg2da9iaa dweadaWgaaqcfasaaiaad6gacaWGJbGaaG4maiabgkHiTiaadMgaca WGZbaabeaajuaGdaqadaqaaiaad6gacaGGSaGaamiBaiaacYcacaaI ZaGaaiilaiaac6cacaGGUaGaaiOlaaGaayjkaiaawMcaaiqbfI6azz aataWaaeWaaeaadaWhdaqaaiqadkhagaWeaaGaayz4GaaacaGLOaGa ayzkaaaaaa@62DA@ (19.2)

In order to use the ordinary product without star product, with new vision, as mentioned before, we apply the Boopp’s shift method on the above eqs. (19.1) and (19.2) to obtain two reduced Schrödinger in both (NC-2D: RSP) and (NC-3D: RSP) for (m.i.s.) potential:

H nc2is ( p ^ i , x ^ i )ψ( r )= E nc2is ( n,l,2,... )ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaGOmaiabgkHiTiaadMgacaWGZbaa beaajuaGdaqadaqaaiqadchagaqcamaaBaaajuaibaGaamyAaaqaba qcfaOaaiilaiqadIhagaqcamaaBaaajuaibaGaamyAaaqabaaajuaG caGLOaGaayzkaaGaeqiYdK3aaeWaaeaaceWGYbGbaSaaaiaawIcaca GLPaaacqGH9aqpcaWGfbWaaSbaaKqbGeaacaWGUbGaam4yaiaaikda cqGHsislcaWGPbGaam4CaaqcfayabaWaaeWaaeaacaWGUbGaaiilai aadYgacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6caaiaawIca caGLPaaacqaHipqEdaqadaqaaiqadkhagaWcaaGaayjkaiaawMcaai aaykW7aaa@601F@ (20.1)

and

H nc3is ( p ^ i , x ^ i )ψ( r )= E nc3is ( n,l,3,... )ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaG4maiabgkHiTiaadMgacaWGZbaa beaajuaGdaqadaqaaiqadchagaqcamaaBaaajuaibaGaamyAaaqaba qcfaOaaiilaiqadIhagaqcamaaBaaajuaqbaGaamyAaaqabaaajuaG caGLOaGaayzkaaGaeqiYdK3aaeWaaeaaceWGYbGbaSaaaiaawIcaca GLPaaacqGH9aqpcaWGfbWaaSbaaKqbGeaacaWGUbGaam4yaiaaioda cqGHsislcaWGPbGaam4CaaqcfayabaWaaeWaaeaacaWGUbGaaiilai aadYgacaGGSaGaaG4maiaacYcacaGGUaGaaiOlaiaac6caaiaawIca caGLPaaacqaHipqEdaqadaqaaiqadkhagaWcaaGaayjkaiaawMcaaa aa@5EB7@ (20.2)

Where the new operators of Hamiltonian H nc2is ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGUbGaam4yaiaaikdacqGHsislcaWGPbGaam4CaaqabaGc daqadaqaaiqadchagaqcamaaBaaaleaacaWGPbaabeaakiaacYcace WG4bGbaKaadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaaa@42F6@ and H nc3is ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaG4maiabgkHiTiaadMgacaWGZbaa juaGbeaadaqadaqaaiqadchagaqcamaaBaaajuaibaGaamyAaaqaba qcfaOaaiilaiqadIhagaqcamaaBaaajuaibaGaamyAaaqcfayabaaa caGLOaGaayzkaaaaaa@456F@ can be expressed in three general varieties: both noncommutative space and noncommutative phase (NC-2D: RSP, NC-3D: RSP), only noncommutative space (NC-2D: RS, NC-3D: RS) and only noncommutative phase (NC: 2D-RP, NC: 3D-RP) as, respectively:

H nc(23)is ( p ^ i , x ^ i )H( p x + θ ¯ 2 y, p y θ ¯ 2 x,x θ 2 p y ,y+ θ 2 p x )    for NC-2D: RSP      and     NC-3D: RSP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaiikaiaaikdacqGHsislcaaIZaGa aiykaiabgkHiTiaadMgacaWGZbaabeaajuaGdaqadaqaaiqadchaga qcamaaBaaajuaibaGaamyAaaqabaqcfaOaaiilaiqadIhagaqcamaa BaaajuaibaGaamyAaaqabaaajuaGcaGLOaGaayzkaaGaeyyyIORaam isamaabmaabaGaamiCamaaBaaajuaibaGaamiEaaqabaqcfaOaey4k aSYaaSaaaeaadaqdaaqaaiabeI7aXbaaaeaacaaIYaaaaiaadMhaca GGSaGaamiCamaaBaaajuaibaGaamyEaaqcfayabaGaeyOeI0YaaSaa aeaadaqdaaqaaiabeI7aXbaaaeaacaaIYaaaaiaadIhacaGGSaGaam iEaiabgkHiTmaalaaabaGaeqiUdehabaGaaGOmaaaacaWGWbWaaSba aKqbGeaacaWG5baajuaGbeaacaGGSaGaamyEaiabgUcaRmaalaaaba GaeqiUdehabaGaaGOmaaaacaWGWbWaaSbaaeaacaWG4baabeaaaiaa wIcacaGLPaaafaqabeqacaaabaGaaeiiaiaabccacaqGGaGaaeOzai aab+gacaqGYbaabaGaaeOtaiaaboeacaqGTaGaaeOmaiaabseacaqG 6aGaaeiiaiaabkfacaqGtbGaaeiuaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabggacaqGUbGaaeizaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeOtaiaaboeacaqGTaGaae4maiaabseacaqG6a GaaeiiaiaabkfacaqGtbGaaeiuaaaaaaa@88A6@ (21.1)

H nc(23)is ( p ^ i , x ^ i )H( p x , p y ,x θ 2 p y ,y+ θ 2 p x )                     for NC-2D: RS       and    NC-3D: RS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaiikaiaaikdacqGHsislcaaIZaGa aiykaiabgkHiTiaadMgacaWGZbaabeaajuaGdaqadaqaaiqadchaga qcamaaBaaajuaibaGaamyAaaqabaqcfaOaaiilaiqadIhagaqcamaa BaaajuaibaGaamyAaaqcfayabaaacaGLOaGaayzkaaGaeyyyIORaam isamaabmaabaGaamiCamaaBaaajuaibaGaamiEaaqabaqcfaOaaiil aiaadchadaWgaaqcfasaaiaadMhaaKqbagqaaiaacYcacaWG4bGaey OeI0YaaSaaaeaacqaH4oqCaeaacaaIYaaaaiaadchadaWgaaqcfasa aiaadMhaaKqbagqaaiaacYcacaWG5bGaey4kaSYaaSaaaeaacqaH4o qCaeaacaaIYaaaaiaadchadaWgaaqaaiaadIhaaeqaaaGaayjkaiaa wMcaauaabeqabiaaaeaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabAgaca qGVbGaaeOCaaqaaiaab6eacaqGdbGaaeylaiaabkdacaqGebGaaeOo aiaabccacaqGsbGaae4uaaaacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaaeiiaiaabccacaqG GaGaaeOtaiaaboeacaqGTaGaae4maiaabseacaqG6aGaaeiiaiaabk facaqGtbaaaa@8840@ (22.2)

H nc(23)is ( p ^ i , x ^ i )H( p x + θ ¯ 2 y, p x θ ¯ 2 x,x,y )                          for NC-2D: RP       and    NC-3D: RP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqaaiaad6gajuaicaWGJbGaaiikaiaaikdacqGHsislcaaIZaGa aiykaiabgkHiTiaadMgacaWGZbaajuaGbeaadaqadaqaaiqadchaga qcamaaBaaajuaibaGaamyAaaqabaqcfaOaaiilaiqadIhagaqcamaa BaaajuaibaGaamyAaaqabaaajuaGcaGLOaGaayzkaaGaeyyyIORaam isamaabmaabaGaamiCamaaBaaajuaibaGaamiEaaqcfayabaGaey4k aSYaaSaaaeaadaqdaaqaaiabeI7aXbaaaeaacaaIYaaaaiaadMhaca GGSaGaamiCamaaBaaajuaibaGaamiEaaqabaqcfaOaeyOeI0YaaSaa aeaadaqdaaqaaiabeI7aXbaaaeaacaaIYaaaaiaadIhacaGGSaGaam iEaiaacYcacaWG5baacaGLOaGaayzkaaqbaeqabeGaaaqaaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGMbGaae4BaiaabkhaaeaacaqGobGaae4qaiaab2cacaqGYaGaaeir aiaabQdacaqGGaGaaeOuaiaabcfaaaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaabccacaqG GaGaaeiiaiaab6eacaqGdbGaaeylaiaabodacaqGebGaaeOoaiaabc cacaqGsbGaaeiuaaaa@88A2@ (22.3)

In recently work, we are interest with the first variety (21.1), after straightforward calculations, we can obtain the five important terms, which will be use to determine the (m.i.s.) potential in (NC: 2D- RSP) and (NC: 3D-RSP), respectively, as:

A r ^ 2 = A r 4 + Aθ L z r 4 ,    B r ^ = B r Bθ L z 2r 3     and      p ^ 2 2 m 0 = p 2 2 m 0 + L θ ¯ 2 m 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaamyqaaqaamaavacabeqabKqbGeaacaaIYaaajuaGbaGabmOCayaa jaaaaaaacqGH9aqpdaWcaaqaaiaadgeaaeaadaqfGaqabeqajuaiba GaaGinaaqcfayaaiaadkhaaaaaaiabgUcaRmaalaaabaGaamyqaiab eI7aXjaadYeadaWgaaqcfasaaiaadQhaaeqaaaqcfayaamaavacabe qabKqbGeaacaaI0aaajuaGbaGaamOCaaaaaaGaaiilaiaabccacaqG GaGaaeiiamaalaaabaGaamOqaaqaaiqadkhagaqcaaaacqGH9aqpda WcaaqaaiaadkeaaeaacaWGYbaaaiabgkHiTmaalaaabaGaamOqaiab eI7aXjaadYeadaWgaaqcfasaaiaadQhaaeqaaaqcfayaamaavacabe qabKqbGeaacaaIZaaajuaGbaGaaGOmaiaadkhaaaaaaiaabccacaqG GaGaaeiiaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiamaalaaabaGabmiCayaajaWaaWbaaeqajuaibaGa aGOmaaaaaKqbagaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfa yabaaaaiabg2da9maalaaabaGaamiCamaaCaaabeqcfasaaiaaikda aaaajuaGbaGaaGOmaiaad2gadaWgaaqcfasaaiaaicdaaeqaaaaaju aGcqGHRaWkdaWcaaqaaGqabiqa=XeagaWcamaaFiaabaWaa0aaaeaa iiqacqGF4oqCaaaacaGLxdcaaeaacaaIYaGaamyBamaaBaaajuaiba GaaGimaaqabaaaaaaa@75A4@ (23)

and

A r ^ 2 = A r 4 + A L Θ r 4 ,    B r ^ = B r B L Θ 2r 3     and     p ^ 2 2 m 0 = p 2 2 m 0 + θ ¯ L z 2 m 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaamyqaaqaamaavacabeqabKqbGeaacaaIYaaajuaGbaGabmOCayaa jaaaaaaacqGH9aqpdaWcaaqaaiaadgeaaeaadaqfGaqabeqajuaiba GaaGinaaqcfayaaiaadkhaaaaaaiabgUcaRmaalaaabaGaamyqaiqa dYeagaWcaiqbfI5arzaalaaabaWaaubiaeqabeqcfasaaiaaisdaaK qbagaacaWGYbaaaaaacaGGSaGaaeiiaiaabccacaqGGaWaaSaaaeaa caWGcbaabaGabmOCayaajaaaaiabg2da9maalaaabaGaamOqaaqaai aadkhaaaGaeyOeI0YaaSaaaeaacaWGcbGabmitayaalaGafuiMdeLb aSaaaeaadaqfGaqabeqajuaibaGaaG4maaqcfayaaiaaikdacaWGYb aaaaaacaqGGaGaaeiiaiaabccacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiaiaabccacaqGGaGaaeiiamaalaaabaGabmiCayaajaWaaWbaae qajuaibaGaaGOmaaaaaKqbagaacaaIYaGaamyBamaaBaaajuaibaGa aGimaaqcfayabaaaaiabg2da9maalaaabaGaamiCamaaCaaajuaibe qaaiaaikdaaaaajuaGbaGaaGOmaiaad2gadaWgaaqcfasaaiaaicda aKqbagqaaaaacqGHRaWkdaWcaaqaamaanaaabaGaeqiUdehaaiaadY eadaWgaaqcfasaaiaadQhaaKqbagqaaaqaaiaaikdacaWGTbWaaSba aKqbGeaacaaIWaaajuaGbeaaaaaaaa@71AC@ (24)

Which allow us to obtaining the global potential operator H nc2is ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaGOmaiabgkHiTiaadMgacaWGZbaa beaajuaGdaqadaqaaiqadchagaqcamaaBaaajuaibaGaamyAaaqaba qcfaOaaiilaiqadIhagaqcamaaBaaajuaibaGaamyAaaqabaaajuaG caGLOaGaayzkaaaaaa@456E@ and H nc3is ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaG4maiabgkHiTiaadMgacaWGZbaa beaajuaGdaqadaqaaiqadchagaqcamaaBaaajuaibaGaamyAaaqaba qcfaOaaiilaiqadIhagaqcamaaBaaajuaibaGaamyAaaqabaaajuaG caGLOaGaayzkaaaaaa@456F@ for (m.i.s) potential in both (NC: 2D-RSP) and (NC: 3D-RSP), respectively, as:

H nc2is ( p ^ i , x ^ i )= A r 2 B r + p 2 2 m 0 + θ ¯ L z 2 m 0 +( A r 4 B 2 r 3 )θ L z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaGOmaiabgkHiTiaadMgacaWGZbaa beaajuaGdaqadaqaaiqadchagaqcamaaBaaajuaibaGaamyAaaqcfa yabaGaaiilaiqadIhagaqcamaaBaaajuaibaGaamyAaaqabaaajuaG caGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaWGbbaabaGaamOCamaaCa aabeqcfasaaiaaikdaaaaaaKqbakabgkHiTmaalaaabaGaamOqaaqa aiaadkhaaaGaey4kaSYaaSaaaeaacaWGWbWaaWbaaeqabaGaaGOmaa aaaeaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfayabaaaaiab gUcaRmaalaaabaWaa0aaaeaacqaH4oqCaaGaamitamaaBaaabaGaam OEaaqabaaabaGaaGOmaiaad2gadaWgaaqcfasaaiaaicdaaKqbagqa aaaacqGHRaWkdaqadaqaamaalaaabaGaamyqaaqaaiaadkhadaahaa qcfasabeaacaaI0aaaaaaajuaGcqGHsisldaWcaaqaaiaadkeaaeaa caaIYaGaamOCamaaCaaajuaibeqaaiaaiodaaaaaaaqcfaOaayjkai aawMcaaiabeI7aXjaadYeadaWgaaqaaiaadQhaaeqaaaaa@691C@ (25.1)

and

H nc3is ( p ^ i , x ^ i )= A r 2 B r + p 2 2 m 0 + L θ ¯ 2 m 0 +( A r 4 B 2 r 3 ) L Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaG4maiabgkHiTiaadMgacaWGZbaa beaajuaGdaqadaqaaiqadchagaqcamaaBaaajuaibaGaamyAaaqcfa yabaGaaiilaiqadIhagaqcamaaBaaajuaibaGaamyAaaqabaaajuaG caGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaWGbbaabaGaamOCamaaCa aajuaibeqaaiaaikdaaaaaaKqbakabgkHiTmaalaaabaGaamOqaaqa aiaadkhaaaGaey4kaSYaaSaaaeaacaWGWbWaaWbaaKqbGeqabaGaaG OmaaaaaKqbagaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfaya baaaaiabgUcaRmaalaaabaacbeGab8htayaalaWaa8Haaeaadaqdaa qaaGGabiab+H7aXbaaaiaawEniaaqaaiaaikdacaWGTbWaaSbaaKqb GeaacaaIWaaabeaaaaqcfaOaey4kaSYaaeWaaeaadaWcaaqaaiaadg eaaeaacaWGYbWaaWbaaKqbGeqabaGaaGinaaaaaaqcfaOaeyOeI0Ya aSaaaeaacaWGcbaabaGaaGOmaiaadkhadaahaaqabKqbGeaacaaIZa aaaaaaaKqbakaawIcacaGLPaaaceWFmbGbaSaacuqHyoqugaWcaaaa @694B@ (25.2)

It’s clearly, that the four first terms are given the ordinary inverse-square potential and kinetic energy in (2D-3D) spaces, while the rest terms are proportional’s with infinitesimals parameters ( , ) and ( , ), thus, we can considered as a perturbations terms, we noted by H ^ 2pert ( r,A,B,θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaGOmaiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeqiUdeNaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaaaa@47F5@ and H ^ 3pert ( r,A,B,Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaG4maiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeuiMdeLaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaaaa@47B7@ for (NC: 2D-RSP) and (NC: 3D-RSP) symmetries, respectively, as:

H ^ 2pert ( r,A,B,θ, θ ¯ )= L z θ ¯ 2 m 0 +( A r 4 B 2 r 3 )θ L z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaGOmaiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaKqbagqaamaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeqiUdeNaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaiabg2da9maalaaabaGaamitamaaBaaajuaibaGaamOEaaqaba qcfa4aa0aaaeaacqaH4oqCaaaabaGaaGOmaiaad2gadaWgaaqcfasa aiaaicdaaKqbagqaaaaacqGHRaWkdaqadaqaamaalaaabaGaamyqaa qaaiaadkhadaahaaqcfasabeaacaaI0aaaaaaajuaGcqGHsisldaWc aaqaaiaadkeaaeaacaaIYaGaamOCamaaCaaajuaibeqaaiaaiodaaa aaaaqcfaOaayjkaiaawMcaaiabeI7aXjaadYeadaWgaaqcfasaaiaa dQhaaKqbagqaaaaa@600D@ (26.1)

and

H ^ 3pert ( r,A,B,Θ, θ ¯ )= L θ ¯ 2 m 0 +( A r 4 B 2 r 3 ) L Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaG4maiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeuiMdeLaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaiabg2da9maalaaabaacbeGab8htayaalaWaa8Haaeaadaqdaa qaaGGabiab+H7aXbaaaiaawEniaaqaaiaaikdacaWGTbWaaSbaaKqb GeaacaaIWaaabeaaaaqcfaOaey4kaSYaaeWaaeaadaWcaaqaaiaadg eaaeaacaWGYbWaaWbaaKqbGeqabaGaaGinaaaaaaqcfaOaeyOeI0Ya aSaaaeaacaWGcbaabaGaaGOmaiaadkhadaahaaqcfasabeaacaaIZa aaaaaaaKqbakaawIcacaGLPaaaceWFmbGbaSaacuqHyoqugaWcaaaa @5DC9@ (26.2)

The Exact Spin-Orbital Hamiltonian and the Corresponding Spectrum for (m.i.s.) Potential in both (NC: 2D- RSP) and (NC: 3D- RSP) Symmetries for Excited States for One-Electron Atoms

The exact spin-orbital hamiltonian for (m.i.s.) potential in both (NC: 2D- RSP) and (NC: 3D- RSP) symmetries for one-electron atoms

 Again, the perturbative two terms H ^ 2pert ( r,A,B,θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaGOmaiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaKqbagqaamaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeqiUdeNaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaaaa@47F5@  and H ^ 3pert ( r,A,B,Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaG4maiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeuiMdeLaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaaaa@47B7@  can be rewritten to the equivalent physical form for (m.i.p.) potential:
H ^ 2pert ( r,A,B,θ, θ ¯ )={ θ ¯ 2 m 0 +θ( A r 4 B 2 r 3 ) } S L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaGOmaiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeqiUdeNaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaiabg2da9maacmaabaWaaSaaaeaadaqdaaqaaiabeI7aXbaaae aacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfayabaaaaiabgUca RiabeI7aXnaabmaabaWaaSaaaeaacaWGbbaabaGaamOCamaaCaaaju aibeqaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGaamOqaaqaaiaa ikdacaWGYbWaaWbaaKqbGeqabaGaaG4maaaaaaaajuaGcaGLOaGaay zkaaaacaGL7bGaayzFaaWaa8XaaeaacaWGtbaacaGLHdcadaWhdaqa aiaadYeaaiaawgoiaaaa@620B@ (26.3)
H ^ 3pert ( r,A,B,Θ, θ ¯ )={ θ ¯ 2 m 0 +Θ( A r 4 B 2 r 3 ) } S L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaG4maiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeuiMdeLaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaiabg2da9maacmaabaWaaSaaaeaadaqdaaqaaiabeI7aXbaaae aacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfayabaaaaiabgUca RiabfI5arnaabmaabaWaaSaaaeaacaWGbbaabaGaamOCamaaCaaaju aibeqaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGaamOqaaqaaiaa ikdacaWGYbWaaWbaaeqajuaibaGaaG4maaaaaaaajuaGcaGLOaGaay zkaaaacaGL7bGaayzFaaWaa8XaaeaacaWGtbaacaGLHdcadaWhdaqa aiaadYeaaiaawgoiaaaa@618E@ (26.4)
Furthermore, the above perturbative terms H ^ 2pert ( r,A,B,θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaGOmaiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeqiUdeNaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaaaa@47F5@  and H ^ 3pert ( r,A,B,Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaG4maiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaKqbagqaamaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeuiMdeLaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaaaa@47B7@  can be rewritten to the following new equivalent form for (m.i.p.) potential:
H ^ 2pert ( r,A,B,θ, θ ¯ )= 1 2 { θ ¯ 2 m 0 +θ( A r 4 B 2 r 3 ) }( J 2 L 2 S 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmisayaaja WaaSbaaSqaaiaaikdacqGHsislcaWGWbGaamyzaiaadkhacaWG0baa beaakmaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadkeacaGGSa GaeqiUdeNaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaawMcaaiab g2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaiWaaeaadaWcaaqaam aanaaabaGaeqiUdehaaaqaaiaaikdacaWGTbWaaSbaaSqaaiaaicda aeqaaaaakiabgUcaRiabeI7aXnaabmaabaWaaSaaaeaacaWGbbaaba GaamOCamaaCaaaleqabaGaaGinaaaaaaGccqGHsisldaWcaaqaaiaa dkeaaeaacaaIYaGaamOCamaaCaaaleqabaGaaG4maaaaaaaakiaawI cacaGLPaaaaiaawUhacaGL9baadaqadaqaamaaFmaabaGaamOsaaGa ayz4GaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Yaa8XaaeaacaWGmb aacaGLHdcadaahaaWcbeqaaiaaikdaaaGccqGHsisldaWhdaqaaiaa dofaaiaawgoiamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaa aa@693D@ (27.1)
H ^ 3pert ( r,A,B,Θ, θ ¯ )= 1 2 { θ ¯ 2 m 0 +Θ( A r 4 B 2 r 3 ) }( J 2 L 2 S 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaaG4maiabgkHiTiaadchacaWGLbGaamOCaiaa dshaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGbbGaaiilaiaadk eacaGGSaGaeuiMdeLaaiilamaanaaabaGaeqiUdehaaaGaayjkaiaa wMcaaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaiWaaeaada WcaaqaamaanaaabaGaeqiUdehaaaqaaiaaikdacaWGTbWaaSbaaKqb GeaacaaIWaaabeaaaaqcfaOaey4kaSIaeuiMde1aaeWaaeaadaWcaa qaaiaadgeaaeaacaWGYbWaaWbaaKqbGeqabaGaaGinaaaaaaqcfaOa eyOeI0YaaSaaaeaacaWGcbaabaGaaGOmaiaadkhadaahaaqcfasabe aacaaIZaaaaaaaaKqbakaawIcacaGLPaaaaiaawUhacaGL9baadaqa daqaamaaFmaabaGaamOsaaGaayz4GaWaaWbaaKqbGeqabaGaaGOmaa aajuaGcqGHsisldaWhdaqaaiaadYeaaiaawgoiamaaCaaajuaibeqa aiaaikdaaaqcfaOaeyOeI0Yaa8XaaeaacaWGtbaacaGLHdcadaahaa qabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaaaa@6DD4@ (27.2)
To the best of our knowledge, we just replace the coupling spin-orbital S L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFmaaba Gaam4uaaGaayz4GaWaa8XaaeaacaWGmbaacaGLHdcaaaa@3BA0@  by the expression 1 2 ( J 2 L 2 S 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaaiaaikdaaaWaaeWaaeaadaWhdaqaaiaadQeaaiaawgoi amaaCaaabeqcfasaaiaaikdaaaqcfaOaeyOeI0Yaa8XaaeaacaWGmb aacaGLHdcadaahaaqcfasabeaacaaIYaaaaKqbakabgkHiTmaaFmaa baGaam4uaaGaayz4GaWaaWbaaKqbGeqabaGaaGOmaaaaaKqbakaawI cacaGLPaaaaaa@47E6@ , in quantum mechanics. The set ( H nc(23)is ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaaiikaiaaikdacqGHsislcaaIZaGa aiykaiabgkHiTiaadMgacaWGZbaabeaajuaGdaqadaqaaiqadchaga qcamaaBaaajuaibaGaamyAaaqabaqcfaOaaiilaiqadIhagaqcamaa BaaajuaibaGaamyAaaqabaaajuaGcaGLOaGaayzkaaaaaa@4871@ , J 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabQeada ahaaqcfasabeaacaqGYaaaaaaa@384B@ , L 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabYeada ahaaqabKqbGeaacaaIYaaaaaaa@3854@ , S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabofada ahaaqabKqbGeaacaaIYaaaaaaa@385B@ and J z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQeada WgaaqcfasaaiaadQhaaKqbagqaaiaacMcaaaa@39D1@  forms a complete of conserved physics quantities and the eigenvalues of the spin orbital coupling operator are:

p ± ( j=l±1/2,l,s=1/2 ) 1 2 { ( l+ 1 2 )(l+ 1 2 +1)+l(l+1) 3 4    p +   for j= l+ 1 2 polarizationup ( l 1 2 )(l 1 2 +1)+l(l+1) 3 4    p    for j= l+ 1 2 polarizationdown MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadchada WgaaqaaiabgglaXcqabaWaaeWaaeaacaWGQbGaeyypa0JaamiBaiab gglaXkaaigdacaGGVaGaaGOmaiaacYcacaWGSbGaaiilaiaadohacq GH9aqpcaaIXaGaai4laiaaikdaaiaawIcacaGLPaaacqGHHjIUdaWc baqaaiaaigdaaeaacaaIYaaaamaaceaaeaqabeaadaqadaqaaiaadY gacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaaGaayjkaiaawMca aiaacIcacaWGSbGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaacq GHRaWkcaaIXaGaaiykaiabgUcaRiaadYgacaGGOaGaamiBaiabgUca RiaaigdacaGGPaGaeyOeI0YaaSaaaeaacaaIZaaabaGaaGinaaaaqa aaaaaaaaWdbiaacckapaGaeyyyIO7dbiaacckapaGaamiCamaaBaaa baGaey4kaScabeaafaqabeqacaaabaGaaeiiaiaabccacaqGMbGaae 4BaiaabkhaaeaacaWGQbGaeyypa0ZdbiaacckapaGaamiBaiabgUca RmaaleaabaGaaGymaaqaaiaaikdaaaGaeyO0H4TaaeiCaiaab+gaca qGSbGaaeyyaiaabkhacaqGPbGaaeOEaiaabggacaqG0bGaaeyAaiaa b+gacaqGUbGaeyOeI0IaaeyDaiaabchaaaaabaWaaeWaaeaacaWGSb GaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaaaiaawIcacaGLPaaa caGGOaGaamiBaiabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaey 4kaSIaaGymaiaacMcacqGHRaWkcaWGSbGaaiikaiaadYgacqGHRaWk caaIXaGaaiykaiabgkHiTmaalaaabaGaaG4maaqaaiaaisdaaaWdbi aacckapaGaeyyyIO7dbiaacckapaGaamiCamaaBaaabaGaeyOeI0ca beaafaqabeqacaaabaGaaeiiaiaabccacaqGGaGaaeOzaiaab+gaca qGYbaabaGaamOAaiabg2da98qacaGGGcWdaiaadYgacqGHRaWkdaWc baqaaiaaigdaaeaacaaIYaaaaiabgkDiElaabchacaqGVbGaaeiBai aabggacaqGYbGaaeyAaiaabQhacaqGHbGaaeiDaiaabMgacaqGVbGa aeOBaiabgkHiTiaabsgacaWGVbGaam4Daiaad6gaaaaaaiaawUhaaa aa@B950@ (27.3)
Which allows us to form a diagonal ( 2×2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaaGOmaiabgEna0kaaikdaaiaawIcacaGLPaaaaaa@3B91@  and ( 3×3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaaG4maiabgEna0kaaiodaaiaawIcacaGLPaaaaaa@3B93@ two matrixes, with non null elements are [ ( H ^ sois ) 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmisayaajaWaaSbaaKqbGeaacaWGZbGaam4BaiabgkHiTiaadMga caWGZbaabeaaaKqbakaawIcacaGLPaaadaWgaaqcfasaaiaaigdaca aIXaaabeaaaaa@4040@ and ( H ^ sois ) 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmisayaajaWaaSbaaKqbGeaacaWGZbGaam4BaiabgkHiTiaadMga caWGZbaajuaGbeaaaiaawIcacaGLPaaadaWgaaqcfasaaiaaikdaca aIYaaajuaGbeaaaaa@40D0@ ] and [ ( H ^ sois ) 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmisayaajaWaaSbaaKqbGeaacaWGZbGaam4BaiabgkHiTiaadMga caWGZbaajuaGbeaaaiaawIcacaGLPaaadaWgaaqcfasaaiaaigdaca aIXaaabeaaaaa@4040@ , ( H ^ sois ) 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmisayaajaWaaSbaaKqbGeaacaWGZbGaam4BaiabgkHiTiaadMga caWGZbaajuaGbeaaaiaawIcacaGLPaaadaWgaaqcfasaaiaaikdaca aIYaaajuaGbeaaaaa@40D0@ , ( H ^ sois ) 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmisayaajaWaaSbaaKqbGeaacaWGZbGaam4BaiabgkHiTiaadMga caWGZbaabeaaaKqbakaawIcacaGLPaaadaWgaaqcfasaaiaaiodaca aIZaaajuaGbeaaaaa@40D2@ ] for (m.i.s.) potential in (NC: 2D-RSP) and (NC: 3D-RSP), respectively, as:

( H soip ) 11 = p + ( θ ¯ 2 m 0 +θ( A r 4 B 2 r 3 ) )if j=l+ 1 2  spin -up ( H soip ) 22 = p ( θ ¯ 2 m 0 +θ( A r 4 B 2 r 3 ) ) if j=l 1 2  spin -down MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aae WaaeaacaWGibWaaSbaaKqbGeaacaWGZbGaam4BaiabgkHiTiaadMga caWGWbaajuaGbeaaaiaawIcacaGLPaaadaWgaaqcfasaaiaaigdaca aIXaaabeaajuaGcqGH9aqpcaWGWbWaaSbaaeaacqGHRaWkaeqaamaa bmaabaWaaSaaaeaadaqdaaqaaiabeI7aXbaaaeaacaaIYaGaamyBam aaBaaajuaibaGaaGimaaqcfayabaaaaiabgUcaRiabeI7aXnaabmaa baWaaSaaaeaacaWGbbaabaGaamOCamaaCaaabeqcfasaaiaaisdaaa aaaKqbakabgkHiTmaalaaabaGaamOqaaqaaiaaikdacaWGYbWaaWba aeqajuaibaGaaG4maaaaaaaajuaGcaGLOaGaayzkaaaacaGLOaGaay zkaaGaaGPaVlaabMgacaqGMbGaaeiiaiaadQgacqGH9aqpcaWGSbGa ey4kaSYaaSqaaeaacaaIXaaabaGaaGOmaaaacaqGGaGaeyO0H4Taae 4CaiaabchacaqGPbGaaeOBaiaabccacaqGTaGaaeyDaiaabchaaOqa aKqbaoaabmaabaGaamisamaaBaaabaqcfaIaam4Caiaad+gacqGHsi slcaWGPbqcfaOaamiCaaqabaaacaGLOaGaayzkaaWaaSbaaKqbGeaa caaIYaGaaGOmaaqabaqcfaOaeyypa0JaamiCamaaBaaabaGaeyOeI0 cabeaadaqadaqaamaalaaabaWaa0aaaeaacqaH4oqCaaaabaGaaGOm aiaad2gadaWgaaqcfasaaiaaicdaaeqaaaaajuaGcqGHRaWkcqaH4o qCdaqadaqaamaalaaabaGaamyqaaqaaiaadkhadaahaaqcfasabeaa caaI0aaaaaaajuaGcqGHsisldaWcaaqaaiaadkeaaeaacaaIYaGaam OCamaaCaaajuaibeqaaiaaiodaaaaaaaqcfaOaayjkaiaawMcaaaGa ayjkaiaawMcaaiaaykW7caqGGaGaaeyAaiaabAgacaqGGaGaamOAai abg2da9iaadYgacqGHsisldaWcbaqaaiaaigdaaeaacaaIYaaaaiaa bccacqGHshI3caqGZbGaaeiCaiaabMgacaqGUbGaaeiiaiaab2caca qGKbGaae4BaiaabEhacaqGUbaaaaa@A15C@ (28.1)

and

( H ^ sois ) 11 = p + { θ ¯ 2 m 0 +Θ( A r 4 B 2 r 3 ) }if j=l+ 1 2  spin up ( H ^ sois ) 22 =  p { θ ¯ 2 m 0 +Θ( A r 4 B 2 r 3 ) } ifj = l 1 2  spin down ( H ^ sois ) 33 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aae WaaeaaceWGibGbaKaadaWgaaqcfasaaiaadohacaWGVbGaeyOeI0Ia amyAaiaadohaaKqbagqaaaGaayjkaiaawMcaamaaBaaajuaibaGaaG ymaiaaigdaaeqaaKqbakabg2da9iaadchadaWgaaqaaiabgUcaRaqa baWaaiWaaeaadaWcaaqaamaanaaabaGaeqiUdehaaaqaaiaaikdaca WGTbWaaSbaaKqbGeaacaaIWaaajuaGbeaaaaGaey4kaSIaeuiMde1a aeWaaeaadaWcaaqaaiaadgeaaeaacaWGYbWaaWbaaKqbGeqabaGaaG inaaaaaaqcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaaGOmaiaadkha daahaaqabKqbGeaacaaIZaaaaaaaaKqbakaawIcacaGLPaaaaiaawU hacaGL9baacaaMc8UaaeyAaiaabAgacaqGGaGaamOAaiabg2da9iaa dYgacqGHRaWkdaWcbaqaaiaaigdaaeaacaaIYaaaaiaabccacqGHsh I3caqGZbGaaeiCaiaabMgacaqGUbGaaeiiaiaabwhacaqGWbaabaWa aeWaaeaaceWGibGbaKaadaWgaaqcfasaaiaadohacaWGVbGaeyOeI0 IaamyAaiaadohaaeqaaaqcfaOaayjkaiaawMcaamaaBaaajuaibaGa aGOmaiaaikdaaeqaaKqbakabg2da9abaaaaaaaaapeGaaiiOa8aaca WGWbWaaSbaaeaacqGHsislaeqaamaacmaabaWaaSaaaeaadaqdaaqa aiabeI7aXbaaaeaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfa yabaaaaiabgUcaRiabfI5arnaabmaabaWaaSaaaeaacaWGbbaabaGa amOCamaaCaaabeqcfasaaiaaisdaaaaaaKqbakabgkHiTmaalaaaba GaamOqaaqaaiaaikdacaWGYbWaaWbaaKqbGeqabaGaaG4maaaaaaaa juaGcaGLOaGaayzkaaaacaGL7bGaayzFaaGaaeiiaiaabMgacaqGMb GaamOAa8qacaGGGcWdaiabg2da98qacaGGGcWdaiaadYgacqGHsisl daWcbaqaaiaaigdaaeaacaaIYaaaaiaabccacqGHshI3caqGZbGaae iCaiaabMgacaqGUbGaaeiiaiaabsgacaqGVbGaae4Daiaab6gaaOqa aKqbaoaabmaabaGabmisayaajaWaaSbaaKqbGeaacaWGZbGaam4Bai abgkHiTiaadMgacaWGZbaabeaaaKqbakaawIcacaGLPaaadaWgaaqc fasaaiaaiodacaaIZaaabeaajuaGcqGH9aqpcaaIWaaaaaa@AEB9@ (28.2)

Substituting two equations (26.1) and (26.2) into two equations (20.1) and (20.12), respectively and then, the radial parts of the modified Schrödinger equations, satisfying the following important two equations:

d 2 R l ( ρ ) d ρ 2 + 2 ρ dR l ( ρ ) dρ +( 1 4 + τ ρ 2A+ l 2 ρ 2 { θ ¯ 2 m 0 +θ( A r 4 B 2 r 3 ) } S L ) R l ( r ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaubeaeqabaGaamiBaaqabeaacaWGKbWaaWbaaKqbGeqabaGaaGOm aaaajuaGcaWGsbaaamaabmaabaGaeqyWdihacaGLOaGaayzkaaaaba Gaamizaiabeg8aYnaaCaaajuaibeqaaiaaikdaaaaaaKqbakabgUca RmaalaaabaGaaGOmaaqaaiabeg8aYbaadaWcaaqaamaavababeqaai aadYgaaeqabaGaamizaiaadkfaaaWaaeWaaeaacqaHbpGCaiaawIca caGLPaaaaeaacaWGKbGaeqyWdihaaiabgUcaRmaabmaabaGaeyOeI0 YaaSaaaeaacaaIXaaabaGaaGinaaaacqGHRaWkdaWcaaqaaiabes8a 0bqaaiabeg8aYbaacqGHsisldaWcaaqaaiaaikdacaWGbbGaey4kaS IaamiBamaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaeqyWdi3aaWba aKqbGeqabaGaaGOmaaaaaaqcfaOaeyOeI0YaaiWaaeaadaWcaaqaam aanaaabaGaeqiUdehaaaqaaiaaikdacaWGTbWaaSbaaKqbGeaacaaI WaaajuaGbeaaaaGaey4kaSIaeqiUde3aaeWaaeaadaWcaaqaaiaadg eaaeaacaWGYbWaaWbaaKqbGeqabaGaaGinaaaaaaqcfaOaeyOeI0Ya aSaaaeaacaWGcbaabaGaaGOmaiaadkhadaahaaqcfasabeaacaaIZa aaaaaaaKqbakaawIcacaGLPaaaaiaawUhacaGL9baadaWhdaqaaiaa dofaaiaawgoiamaaFmaabaGaamitaaGaayz4GaaacaGLOaGaayzkaa WaaubeaeqabaGaamiBaaqabeaacaWGsbaaamaabmaabaGaamOCaaGa ayjkaiaawMcaaabaaaaaaaaapeGaaiiOa8aacqGH9aqppeGaaiiOa8 aacaaIWaaaaa@8444@ (29.1)

and

1 r 2 r ( r 2 r ) R l ( r )+[ 2( E nc3is ( n,l,3,... )V( r ) ) l( l+1 ) r 2 { θ ¯ 2 m 0 +Θ( A r 4 B 2 r 3 ) } S L ] R l ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaamaavacabeqabKqbGeaacaaIYaaajuaGbaGaamOCaaaa aaWaaSaaaeaacqGHciITaeaacqGHciITcaWGYbaaamaabmaabaWaau biaeqabeqcfasaaiaaikdaaKqbagaacaWGYbaaamaalaaabaGaeyOa IylabaGaeyOaIyRaamOCaaaaaiaawIcacaGLPaaadaqfqaqabeaaca WGSbaabeqaaiaadkfaaaWaaeWaaeaacaWGYbaacaGLOaGaayzkaaGa ey4kaSYaamWaaeaacaaIYaWaaeWaaeaacaWGfbWaaSbaaKqbGeaaca WGUbGaam4yaiaaiodacqGHsislcaWGPbGaam4CaaqcfayabaWaaeWa aeaacaWGUbGaaiilaiaadYgacaGGSaGaaG4maiaacYcacaGGUaGaai Olaiaac6caaiaawIcacaGLPaaacqGHsislcaWGwbWaaeWaaeaacaWG YbaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaeyOeI0YaaSaaaeaaca WGSbWaaeWaaeaacaWGSbGaey4kaSIaaGymaaGaayjkaiaawMcaaaqa amaavacabeqabKqbGeaacaaIYaaajuaGbaGaamOCaaaaaaGaeyOeI0 YaaiWaaeaadaWcaaqaamaanaaabaGaeqiUdehaaaqaaiaaikdacaWG TbWaaSbaaKqbGeaacaaIWaaabeaaaaqcfaOaey4kaSIaeuiMde1aae WaaeaadaWcaaqaaiaadgeaaeaacaWGYbWaaWbaaKqbGeqabaGaaGin aaaaaaqcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaaGOmaiaadkhada ahaaqabKqbGeaacaaIZaaaaaaaaKqbakaawIcacaGLPaaaaiaawUha caGL9baadaWhdaqaaiaadofaaiaawgoiamaaFmaabaGaamitaaGaay z4GaaacaGLBbGaayzxaaWaaubeaeqabaGaamiBaaqabeaacaWGsbaa amaabmaabaGaamOCaaGaayjkaiaawMcaaiabg2da9iaaicdaaaa@8BAC@ (29.2)

for (m.i.s.) potential in (NC: 2D-RSP) and (NC: 3D-RSP), ii is clearly that the above equations including equations (26.1) and (26.2), the perturbative terms of Hamiltonian operator, which we are subject of discussion in next sub-section.

The exact spin-orbital spectrum for (m.i.s.) potential in both (NC: 2D- RSP) and (NC: 3D- RSP) symmetries for states for one-electron atoms

 In this sub section, we are going to study the modifications to the energy levels ( E ncper:u ( θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeqiUdeNaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@4533@ , E ncper:D ( θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadseaaeqaaKqbaoaabmaabaGaeqiUdeNaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@4502@ ) and ( E ncper:u ( Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@44F4@ , E ncper:D ( Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadseaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@44C3@ ) for spin up and spin down, respectively, at first order of parameters ( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ , θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaraaaaa@3847@ ) and ( Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI5arb aa@37F0@ , θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaraaaaa@3847@ ), for excited states n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada ahaaqabKqbGeaacaWG0bGaamiAaaaaaaa@39A2@ , obtained by applying the standard perturbation theory, using eqs. (14.1) (14.2), (27.1) and (27.2) corresponding (NC-2D: RSP) and (NC-3D: RSP), respectively, as:
E ncper:u ( θ, θ ¯ )2 p + R * ( r )[ θ( A r 4 B 2r 3 )+ θ ¯ 2 m 0 ]R( r )rdr        Si j=l+ 1 2 E ncper:D ( θ, θ ¯ )2 p R * ( r )[ θ( A r 4 B 2r 3 )+ θ ¯ 2 m 0 ]R( r )rdr        Si j=l 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGWbGaamyzaiaa dkhacaGG6aGaamyDaaqabaqcfa4aaeWaaeaacqaH4oqCcaGGSaGafq iUdeNbaebaaiaawIcacaGLPaaacqGHHjIUcaaIYaGaey4dIuTaamiC amaaBaaabaGaey4kaScabeaadaWdbaqaaiaadkfadaahaaqcfasabe aacaGGQaaaaKqbaoaabmaabaGaamOCaaGaayjkaiaawMcaamaadmaa baGaeqiUde3aaeWaaeaadaWcaaqaaiaadgeaaeaadaqfGaqabeqaju aibaGaaGinaaqcfayaaiaadkhaaaaaaiabgkHiTmaalaaabaGaamOq aaqaamaavacabeqabKqbGeaacaaIZaaajuaGbaGaaGOmaiaadkhaaa aaaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGafqiUdeNbaebaaeaa caaIYaGaamyBamaaBaaajuaibaGaaGimaaqabaaaaaqcfaOaay5wai aaw2faaiaadkfadaqadaqaaiaadkhaaiaawIcacaGLPaaacaWGYbGa amizaiaadkhaaeqabeGaey4kIipafaqabeqacaaabaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaadofacaWGPbaabaGa amOAaiabg2da9iaadYgacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYa aaaaaaaOqaaKqbakaadweadaWgaaqcfasaaiaad6gacaWGJbGaeyOe I0IaamiCaiaadwgacaWGYbGaaiOoaiaadseaaeqaaKqbaoaabmaaba GaeqiUdeNaaiilaiqbeI7aXzaaraaacaGLOaGaayzkaaGaeyyyIORa aGOmaiabg+GivlaadchadaWgaaqaaiabgkHiTaqabaWaa8qaaeaaca WGsbWaaWbaaKqbGeqabaGaaiOkaaaajuaGdaqadaqaaiaadkhaaiaa wIcacaGLPaaadaWadaqaaiabeI7aXnaabmaabaWaaSaaaeaacaWGbb aabaWaaubiaeqabeqcfasaaiaaisdaaKqbagaacaWGYbaaaaaacqGH sisldaWcaaqaaiaadkeaaeaadaqfGaqabeqajuaibaGaaG4maaqcfa yaaiaaikdacaWGYbaaaaaaaiaawIcacaGLPaaacqGHRaWkdaWcaaqa aiqbeI7aXzaaraaabaGaaGOmaiaad2gadaWgaaqcfasaaiaaicdaaK qbagqaaaaaaiaawUfacaGLDbaacaWGsbWaaeWaaeaacaWGYbaacaGL OaGaayzkaaGaamOCaiaadsgacaWGYbaabeqabiabgUIiYdqbaeqabe GaaaqaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caWGtbGaamyAaaqaaiaadQgacqGH9aqpcaWGSbGaeyOeI0YaaSaaae aacaaIXaaabaGaaGOmaaaaaaaaaaa@B977@  (30.1)
and

E ncper:u ( Θ, θ ¯ ) α p + ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ρ 2l+2 e ρ [ L nl1 2l+1 ( ρ ) ] 2 [ Θ( A' ρ 4 B' 2ρ 3 )+ θ ¯ 2 m 0 ]dρ E ncper:D ( Θ, θ ¯ ) α p ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ρ 2l+2 e ρ [ L nl1 2l+1 ( ρ ) ] 2 [ Θ( A' ρ 4 B' 2ρ 3 )+ θ ¯ 2 m 0 ]dρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGWbGaamyzaiaa dkhacaGG6aGaamyDaaqabaqcfa4aaeWaaeaacqqHyoqucaGGSaGafq iUdeNbaebaaiaawIcacaGLPaaacqGHHjIUcqGHsisldaWcaaqaaiab eg7aHjaadchadaWgaaqaaiabgUcaRaqabaaabaWaaeWaaeaacqGHsi slcaaI4aGaamyraaGaayjkaiaawMcaamaaCaaajuaibeqaaiaaioda caGGVaGaaGOmaaaaaaqcfa4aaeWaaeaadaWcaaqaaiaaikdacaWGcb aabaGaamOBaaaaaiaawIcacaGLPaaadaahaaqcfasabeaacaaIZaaa aKqbaoaalaaabaWaaeWaaeaacaWGUbGaeyOeI0IaamiBaiabgkHiTi aaigdaaiaawIcacaGLPaaacaGGHaaabaGaaGOmaiaad6gadaqadaqa aiaad6gacqGHRaWkcaWGSbaacaGLOaGaayzkaaGaaiyiaaaadaWdba qaamaavacabeqabKqbGeaacaaIYaGaamiBaiabgUcaRiaaikdaaKqb agaacqaHbpGCaaWaaubiaeqabeqcfasaaiabgkHiTiabeg8aYbqcfa yaaiaadwgaaaWaamWaaeaadaqfWaqabKqbGeaacaWGUbGaeyOeI0Ia amiBaiabgkHiTiaaigdaaeaacaaIYaGaamiBaiabgUcaRiaaigdaaK qbagaacaWGmbaaamaabmaabaGaeqyWdihacaGLOaGaayzkaaaacaGL BbGaayzxaaWaaWbaaeqajuaibaGaaGOmaaaajuaGdaWadaqaaiabfI 5arnaabmaabaWaaSaaaeaacaWGbbGaai4jaaqaamaavacabeqabKqb GeaacaaI0aaajuaGbaGaeqyWdihaaaaacqGHsisldaWcaaqaaiaadk eacaGGNaaabaWaaubiaeqabeqcfasaaiaaiodaaKqbagaacaaIYaGa eqyWdihaaaaaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiqbeI7aXz aaraaabaGaaGOmaiaad2gadaWgaaqcfasaaiaaicdaaKqbagqaaaaa aiaawUfacaGLDbaacaWGKbGaeqyWdihabeqabiabgUIiYdaakeaaju aGcaWGfbWaaSbaaKqbGeaacaWGUbGaam4yaiabgkHiTiaadchacaWG LbGaamOCaiaacQdacaWGebaabeaajuaGdaqadaqaaiabfI5arjaacY cacuaH4oqCgaqeaaGaayjkaiaawMcaaiabggMi6kabgkHiTmaalaaa baGaeqySdeMaamiCamaaBaaabaGaeyOeI0cabeaaaeaadaqadaqaai abgkHiTiaaiIdacaWGfbaacaGLOaGaayzkaaWaaWbaaKqbGeqabaGa aG4maiaac+cacaaIYaaaaaaajuaGdaqadaqaamaalaaabaGaaGOmai aadkeaaeaacaWGUbaaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaa iodaaaqcfa4aaSaaaeaadaqadaqaaiaad6gacqGHsislcaWGSbGaey OeI0IaaGymaaGaayjkaiaawMcaaiaacgcaaeaacaaIYaGaamOBamaa bmaabaGaamOBaiabgUcaRiaadYgaaiaawIcacaGLPaaacaGGHaaaam aapeaabaWaaubiaeqabeqcfasaaiaaikdacaWGSbGaey4kaSIaaGOm aaqcfayaaiabeg8aYbaadaqfGaqabeqajuaibaGaeyOeI0IaeqyWdi hajuaGbaGaamyzaaaadaWadaqaamaavadajuaibeqaaiaad6gacqGH sislcaWGSbGaeyOeI0IaaGymaaqaaiaaikdacaWGSbGaey4kaSIaaG ymaaqcfayaaiaadYeaaaWaaeWaaeaacqaHbpGCaiaawIcacaGLPaaa aiaawUfacaGLDbaadaahaaqcfasabeaacaaIYaaaaKqbaoaadmaaba GaeuiMde1aaeWaaeaadaWcaaqaaiaadgeacaGGNaaabaWaaubiaeqa beqcfasaaiaaisdaaKqbagaacqaHbpGCaaaaaiabgkHiTmaalaaaba GaamOqaiaacEcaaeaadaqfGaqabeqajuaibaGaaG4maaqcfayaaiaa ikdacqaHbpGCaaaaaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGafq iUdeNbaebaaeaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfaya baaaaaGaay5waiaaw2faaiaadsgacqaHbpGCaeqabeGaey4kIipaaa aa@010E@ (30.2)

A direct simplification gives:

E ncper:u ( θ, θ ¯ ) 2 p + ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( θ i=1 2 T i2 + θ ¯ 2 m 0 T 32 ) E ncper:D ( θ, θ ¯ ) 2 p ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( θ i=1 2 T i2 + θ ¯ 2 m 0 T 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGWbGaamyzaiaa dkhacaGG6aGaamyDaaqabaqcfa4aaeWaaeaacqaH4oqCcaGGSaGafq iUdeNbaebaaiaawIcacaGLPaaacqGHHjIUdaWcaaqaaiaaikdacqGH pis1caWGWbWaaSbaaeaacqGHRaWkaeqaaaqaamaabmaabaGaeyOeI0 IaaGioaiaadweaaiaawIcacaGLPaaaaaWaaeWaaeaadaWcaaqaaiaa isdacqqHsoGqaeaacaaIYaGaamOBaiabgkHiTiaaikdacaWGTbGaey 4kaSIaaGOmaiaadohadaWgaaqaaiaaikdaaeqaaiabgkHiTiaaigda aaaacaGLOaGaayzkaaWaaWbaaKqbGeqabaGaaGOmaaaajuaGdaqada qaamaalaaabaWaaeWaaeaacaWGUbGaeyOeI0IaamyBaiabgkHiTiaa igdaaiaawIcacaGLPaaacaGGHaaabaWaaeWaaeaacaaIYaGaamOBai abgkHiTiaaikdacaWGTbGaey4kaSIaaGOmaiaadohadaWgaaqcfasa aiaaikdaaKqbagqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaqada qaaiaad6gacqGHsislcaWGTbGaey4kaSIaaGOmaiaadohadaWgaaqc fasaaiaaikdaaKqbagqaaiabgkHiTiaaigdaaiaawIcacaGLPaaaca GGHaaaaaGaayjkaiaawMcaamaabmaabaGaeqiUde3aaabCaeaacaWG ubWaaSbaaeaacaWGPbGaeyOeI0IaaGOmaaqabaaajuaibaGaamyAai abg2da9iaaigdaaeaacaaIYaaajuaGcqGHris5aiabgUcaRmaalaaa baGafqiUdeNbaebaaeaacaaIYaWaaubeaeqajuaibaGaaGimaaqcfa yabeaacaWGTbaaaaaacaWGubWaaSbaaKqbGeaacaaIZaGaeyOeI0Ia aGOmaaqcfayabaaacaGLOaGaayzkaaaakeaajuaGcaWGfbWaaSbaaK qbGeaacaWGUbGaam4yaiabgkHiTiaadchacaWGLbGaamOCaiaacQda caWGebaajuaGbeaadaqadaqaaiabeI7aXjaacYcacuaH4oqCgaqeaa GaayjkaiaawMcaaiabggMi6oaalaaabaGaaGOmaiabg+Givlaadcha daWgaaqaaiabgkHiTaqabaaabaWaaeWaaeaacqGHsislcaaI4aGaam yraaGaayjkaiaawMcaaaaadaqadaqaamaalaaabaGaaGinaiabfk5a cbqaaiaaikdacaWGUbGaeyOeI0IaaGOmaiaad2gacqGHRaWkcaaIYa Gaam4CamaaBaaajuaibaGaaGOmaaqcfayabaGaeyOeI0IaaGymaaaa aiaawIcacaGLPaaadaahaaqcfasabeaacaaIYaaaaKqbaoaabmaaba WaaSaaaeaadaqadaqaaiaad6gacqGHsislcaWGTbGaeyOeI0IaaGym aaGaayjkaiaawMcaaiaacgcaaeaadaqadaqaaiaaikdacaWGUbGaey OeI0IaaGOmaiaad2gacqGHRaWkcaaIYaGaam4CamaaBaaajuaibaGa aGOmaaqabaqcfaOaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaaba GaamOBaiabgkHiTiaad2gacqGHRaWkcaaIYaGaam4CamaaBaaajuai baGaaGOmaaqabaqcfaOaeyOeI0IaaGymaaGaayjkaiaawMcaaiaacg caaaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCdaaeWbqaaiaadsfa daWgaaqcfasaaiaadMgacqGHsislcaaIYaaabeaaaeaacaWGPbGaey ypa0JaaGymaaqaaiaaikdaaKqbakabggHiLdGaey4kaSYaaSaaaeaa cuaH4oqCgaqeaaqaaiaaikdadaqfqaqabKqbGeaacaaIWaaajuaGbe qaaiaad2gaaaaaaiaadsfadaWgaaqcfasaaiaaiodacqGHsislcaaI YaaajuaGbeaaaiaawIcacaGLPaaaaaaa@F2F6@ (31.1)

and

E ncper:u ( Θ, θ ¯ ) α p + ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ( Θ i=1 2 T i3 + θ ¯ 2 m 0 T 33 ) E ncper:D ( Θ, θ ¯ ) α p ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ( Θ i=1 2 T i3 + θ ¯ 2 m 0 T 33 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGWbGaamyzaiaa dkhacaGG6aGaamyDaaqabaqcfa4aaeWaaeaacqqHyoqucaGGSaGafq iUdeNbaebaaiaawIcacaGLPaaacqGHHjIUcqGHsisldaWcaaqaaiab eg7aHjaadchadaWgaaqaaiabgUcaRaqabaaabaWaaeWaaeaacqGHsi slcaaI4aGaamyraaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaioda caGGVaGaaGOmaaaaaaqcfa4aaeWaaeaadaWcaaqaaiaaikdacaWGcb aabaGaamOBaaaaaiaawIcacaGLPaaadaahaaqcfasabeaacaaIZaaa aKqbaoaalaaabaWaaeWaaeaacaWGUbGaeyOeI0IaamiBaiabgkHiTi aaigdaaiaawIcacaGLPaaacaGGHaaabaGaaGOmaiaad6gadaqadaqa aiaad6gacqGHRaWkcaWGSbaacaGLOaGaayzkaaGaaiyiaaaadaqada qaaiabfI5arnaaqahabaGaamivamaaBaaajuaibaGaamyAaiabgkHi TiaaiodaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaaGOmaaqcfa OaeyyeIuoacqGHRaWkdaWcaaqaaiqbeI7aXzaaraaabaGaaGOmamaa vababeqcfasaaiaaicdaaKqbagqabaGaamyBaaaaaaGaamivamaaBa aajuaibaGaaG4maiabgkHiTiaaiodaaeqaaaqcfaOaayjkaiaawMca aaGcbaqcfaOaamyramaaBaaajuaibaGaamOBaiaadogacqGHsislca WGWbGaamyzaiaadkhacaGG6aGaamiraaqabaqcfa4aaeWaaeaacqqH yoqucaGGSaGafqiUdeNbaebaaiaawIcacaGLPaaacqGHHjIUcqGHsi sldaWcaaqaaiabeg7aHjaadchadaWgaaqaaiabgkHiTaqabaaabaWa aeWaaeaacqGHsislcaaI4aGaamyraaGaayjkaiaawMcaamaaCaaabe qcfasaaiaaiodacaGGVaGaaGOmaaaaaaqcfa4aaeWaaeaadaWcaaqa aiaaikdacaWGcbaabaGaamOBaaaaaiaawIcacaGLPaaadaahaaqcfa sabeaacaaIZaaaaKqbaoaalaaabaWaaeWaaeaacaWGUbGaeyOeI0Ia amiBaiabgkHiTiaaigdaaiaawIcacaGLPaaacaGGHaaabaGaaGOmai aad6gadaqadaqaaiaad6gacqGHRaWkcaWGSbaacaGLOaGaayzkaaGa aiyiaaaadaqadaqaaiabfI5arnaaqahabaGaamivamaaBaaajuaiba GaamyAaiabgkHiTiaaiodaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaa baGaaGOmaaqcfaOaeyyeIuoacqGHRaWkdaWcaaqaaiqbeI7aXzaara aabaGaaGOmamaavababeqcfasaaiaaicdaaKqbagqabaGaamyBaaaa aaGaamivamaaBaaajuaibaGaaG4maiabgkHiTiaaiodaaKqbagqaaa GaayjkaiaawMcaaaaaaa@C2E6@ (32.2)

Where, the 6- terms: ( T i2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaadMgacqGHsislcaaIYaaabeaaaaa@3A38@ , T i3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaadMgacqGHsislcaaIZaaajuaGbeaaaaa@3AC7@   i=1,2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaIXaGaaiilaiaaikdaaaa@3A94@ ), T 32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaiodacqGHsislcaaIYaaajuaGbeaaaaa@3A95@  and T 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaiodacqGHsislcaaIZaaajuaGbeaaaaa@3A96@  are given by:

T 12= A ' 0 + e ρ ρ 2 S 2 3 [ L nm1 2 S 2 ( ρ ) ] 2 dρ T 22= B 2 ' 0 + e ρ ρ 2 S 2 2 [ L nm1 2 S 2 ( ρ ) ] 2 dρ T 32= 0 + e ρ ρ 2 S 2 +1 [ L nm1 2 S 2 ( ρ ) ] 2 dρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam ivamaaBaaabaqcfaIaaGymaiabgkHiTiaaikdajuaGcqGH9aqpcaWG bbWaaWbaaeqabaGaai4jaaaaaeqaamaapehabaGaamyzamaaCaaaju aibeqaaiabgkHiTiabeg8aYbaaaeaacaaIWaaabaGaey4kaSIaeyOh IukajuaGcqGHRiI8aiabeg8aYnaaCaaabeqcfasaaiaaikdacaWGtb qcfa4aaSbaaKqbGeaacaaIYaqcfa4aaWbaaKqbGeqabaGaeyOeI0Ia aG4maaaaaeqaaaaajuaGdaWadaqaaiaadYeadaqhaaqcfasaaiaad6 gacqGHsislcaWGTbGaeyOeI0IaaGymaaqaaiaaikdacaWGtbqcfa4a aSbaaKqbGeaacaaIYaaabeaaaaqcfa4aaeWaaeaacqaHbpGCaiaawI cacaGLPaaaaiaawUfacaGLDbaadaahaaqcfasabeaacaaIYaaaaKqb akaadsgacqaHbpGCaeaacaWGubWaaSbaaeaajuaicaaIYaGaeyOeI0 IaaGOmaKqbakabg2da9maalaaabaGaamOqaaqaaiaaikdaaaWaaWba aeqabaGaai4jaaaaaeqaamaapehabaGaamyzamaaCaaabeqcfasaai abgkHiTiabeg8aYbaaaeaacaaIWaaabaGaey4kaSIaeyOhIukajuaG cqGHRiI8aiabeg8aYnaaCaaabeqcfasaaiaaikdacaWGtbqcfa4aaS baaKqbGeaacaaIYaqcfa4aaWbaaKqbGeqabaGaeyOeI0IaaGOmaaaa aeqaaaaajuaGdaWadaqaaiaadYeadaqhaaqcfasaaiaad6gacqGHsi slcaWGTbGaeyOeI0IaaGymaaqaaiaaikdacaWGtbqcfa4aaSbaaKqb GeaacaaIYaaabeaaaaqcfa4aaeWaaeaacqaHbpGCaiaawIcacaGLPa aaaiaawUfacaGLDbaadaahaaqcfasabeaacaaIYaaaaKqbakaadsga cqaHbpGCaeaacaWGubWaaSbaaKqbGeaacaaIZaGaeyOeI0IaaGOmaK qbakabg2da9aqabaWaa8qCaeaacaWGLbWaaWbaaKqbGeqabaGaeyOe I0IaeqyWdihaaaqaaiaaicdaaeaacqGHRaWkcqGHEisPaKqbakabgU IiYdGaeqyWdi3aaWbaaeqajuaibaGaaGOmaiaadofajuaGdaWgaaqc fasaaiaaikdajuaGdaahaaqcfasabeaacqGHRaWkcaaIXaaaaaqaba aaaKqbaoaadmaabaGaamitamaaDaaajuaibaGaamOBaiabgkHiTiaa d2gacqGHsislcaaIXaaabaGaaGOmaiaadofajuaGdaWgaaqcfasaai aaikdaaeqaaaaajuaGdaqadaqaaiabeg8aYbGaayjkaiaawMcaaaGa ay5waiaaw2faamaaCaaajuaibeqaaiaaikdaaaqcfaOaamizaiabeg 8aYbaaaa@B8B7@ (33.1)

T 13= A ' 0 + ρ 2l2 e ρ [ L nl1 2l+1 ( ρ ) ] 2 dρ T 23= B 2 ' 0 + ρ 2l1 e ρ [ L nl1 2l+1 ( ρ ) ] 2 dρ T 33= 0 + ρ 2l+2 e ρ [ L nl1 2l+1 ( ρ ) ] 2 dρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam ivamaaBaaabaqcfaIaaGymaiabgkHiTiaaiodajuaGcqGH9aqpcaWG bbWaaWbaaeqabaGaai4jaaaaaeqaamaapehabaGaeqyWdi3aaWbaaK qbGeqabaGaaGOmaiaadYgacqGHsislcaaIYaaaaaqaaiaaicdaaeaa cqGHRaWkcqGHEisPaKqbakabgUIiYdGaamyzamaaCaaajuaibeqaai abgkHiTiabeg8aYbaajuaGdaWadaqaaiaadYeadaqhaaqcfasaaiaa d6gacqGHsislcaWGSbGaeyOeI0IaaGymaaqaaiaaikdacaWGSbGaey 4kaSIaaGymaaaajuaGdaqadaqaaiabeg8aYbGaayjkaiaawMcaaaGa ay5waiaaw2faamaaCaaajuaibeqaaiaaikdaaaqcfaOaamizaiabeg 8aYbqaaiaadsfadaWgaaqaaKqbGiaaikdacqGHsislcaaIZaqcfaOa eyypa0ZaaSaaaeaacaWGcbaabaGaaGOmaaaadaahaaqabeaacaGGNa aaaaqabaWaa8qCaeaacqaHbpGCdaahaaqcfasabeaacaaIYaGaamiB aiabgkHiTiaaigdaaaaabaGaaGimaaqaaiabgUcaRiabg6HiLcqcfa Oaey4kIipacaWGLbWaaWbaaKqbGeqabaGaeyOeI0IaeqyWdihaaKqb aoaadmaabaGaamitamaaDaaajuaibaGaamOBaiabgkHiTiaadYgacq GHsislcaaIXaaabaGaaGOmaiaadYgacqGHRaWkcaaIXaaaaKqbaoaa bmaabaGaeqyWdihacaGLOaGaayzkaaaacaGLBbGaayzxaaWaaWbaaK qbGeqabaGaaGOmaaaajuaGcaWGKbGaeqyWdihabaGaamivamaaBaaa juaibaGaaG4maiabgkHiTiaaiodajuaGcqGH9aqpaeqaamaapehaba GaeqyWdi3aaWbaaKqbGeqabaGaaGOmaiaadYgacqGHRaWkcaaIYaaa aaqaaiaaicdaaeaacqGHRaWkcqGHEisPaKqbakabgUIiYdGaamyzam aaCaaajuaibeqaaiabgkHiTiabeg8aYbaajuaGdaWadaqaaiaadYea daqhaaqcfasaaiaad6gacqGHsislcaWGSbGaeyOeI0IaaGymaaqaai aaikdacaWGSbGaey4kaSIaaGymaaaajuaGdaqadaqaaiabeg8aYbGa ayjkaiaawMcaaaGaay5waiaaw2faamaaCaaajuaibeqaaiaaikdaaa qcfaOaamizaiabeg8aYbaaaa@B1F3@ (33.2)

With new notation A'= ( 8E ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca GGNaGaeyypa0ZaaeWaaeaacqGHsislcaaI4aGaamyraaGaayjkaiaa wMcaamaaCaaabeqcfasaaiaaikdaaaaaaa@3DFE@  and B'= ( 8E ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkeaca GGNaGaeyypa0ZaaeWaaeaacqGHsislcaaI4aGaamyraaGaayjkaiaa wMcaamaaCaaabeqcfasaaiaaiodacaGGVaGaaGOmaaaaaaa@3F6F@ , know we apply the special integral [1, 61]:

J n,α ( γ ) = 0 e x x α+γ [ L n α ( x ) ] 2 dx= ( α+n )! n! k=0 n (1)k Γ( n+κ+γ ) Γ( κγ ) ( α+k+γ )! ( α+k )! 1 κ!( nκ )! , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavadabe qaaKqbGiaad6gajuaGcaGGSaGaeqySdegajuaibaqcfa4aaeWaaKqb GeaacqaHZoWzaiaawIcacaGLPaaaaKqbagaacaWGkbaaaiabg2da9m aapehabaWaaubiaeqabeqcfasaaiabgkHiTiaadIhaaKqbagaacaWG LbaaaaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYdWaaubiae qabeqcfasaaiabeg7aHjabgUcaRiabeo7aNbqcfayaaiaadIhaaaWa aubiaeqabeqcfasaaiaaikdaaKqbagaadaWadaqaamaavadabeqcfa saaiaad6gaaeaacqaHXoqyaKqbagaacaWGmbaaamaabmaabaGaamiE aaGaayjkaiaawMcaaaGaay5waiaaw2faaaaacaWGKbGaamiEaiabg2 da9maalaaabaWaaeWaaeaacqaHXoqycqGHRaWkcaWGUbaacaGLOaGa ayzkaaGaaiyiaaqaaiaad6gacaGGHaaaamaaqahabaGaaiikaiabgk HiTiaaigdacaGGPaGaam4AamaalaaabaGaeu4KdC0aaeWaaeaacaWG UbGaey4kaSIaeqOUdSMaey4kaSIaeq4SdCgacaGLOaGaayzkaaaaba Gaeu4KdC0aaeWaaeaacqGHsislcqaH6oWAcqGHsislcqaHZoWzaiaa wIcacaGLPaaaaaaajuaibaGaam4Aaiabg2da9iaaicdaaeaacaWGUb aajuaGcqGHris5amaalaaabaWaaeWaaeaacqaHXoqycqGHRaWkcaWG RbGaey4kaSIaeq4SdCgacaGLOaGaayzkaaGaaiyiaaqaamaabmaaba GaeqySdeMaey4kaSIaam4AaaGaayjkaiaawMcaaiaacgcaaaWaaSaa aeaacaaIXaaabaGaeqOUdSMaaiyiamaabmaabaGaamOBaiabgkHiTi abeQ7aRbGaayjkaiaawMcaaiaacgcaaaGaaiilaaaa@984E@ (34)

Re( α+γ+1 )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGackfaca GGLbWaaeWaaeaacqaHXoqycqGHRaWkcqaHZoWzcqGHRaWkcaaIXaaa caGLOaGaayzkaaGaeyOkJeVaaGimaaaa@420C@ ,  can be takes: (-3, -2 and +1), α=2 s 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHj abg2da9iaaikdacaWGZbWaaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@3C6B@  and nnm1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gacq GHsgIRcaWGUbGaeyOeI0IaamyBaiabgkHiTiaaigdaaaa@3DD3@ , which allow us to obtaining in (NC: 2D-RSP):

T 12 =A' J nm1,2l+1 ( 3 ) = ( 2 s 2 +nm1 )! ( nm1 )! k=0 n (1)k Γ( nm+κ4 ) Γ( κ+3 ) ( 2 s 2 +k3 )! ( 2 s 2 +k )! 1 κ!( nm1κ )! , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaigdacqGHsislcaaIYaaajuaGbeaacqGH9aqpcaWG bbGaai4jamaavadajuaibeqaaiaad6gacqGHsislcaWGTbGaeyOeI0 IaaGymaiaacYcacaaIYaGaamiBaiabgUcaRiaaigdaaeaajuaGdaqa daqcfasaaiabgkHiTiaaiodaaiaawIcacaGLPaaaaKqbagaacaWGkb aaaiabg2da9maalaaabaWaaeWaaeaacaaIYaGaam4CamaaBaaabaGa aGOmaaqabaGaey4kaSIaamOBaiabgkHiTiaad2gacqGHsislcaaIXa aacaGLOaGaayzkaaGaaiyiaaqaamaabmaabaGaamOBaiabgkHiTiaa d2gacqGHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaaadaaeWbqaai aacIcacqGHsislcaaIXaGaaiykaiaadUgadaWcaaqaaiabfo5ahnaa bmaabaGaamOBaiabgkHiTiaad2gacqGHRaWkcqaH6oWAcqGHsislca aI0aaacaGLOaGaayzkaaaabaGaeu4KdC0aaeWaaeaacqGHsislcqaH 6oWAcqGHRaWkcaaIZaaacaGLOaGaayzkaaaaaaqcfasaaiaadUgacq GH9aqpcaaIWaaabaGaamOBaaqcfaOaeyyeIuoadaWcaaqaamaabmaa baGaaGOmaiaadohadaWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRi aadUgacqGHsislcaaIZaaacaGLOaGaayzkaaGaaiyiaaqaamaabmaa baGaaGOmaiaadohadaWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRi aadUgaaiaawIcacaGLPaaacaGGHaaaamaalaaabaGaaGymaaqaaiab eQ7aRjaacgcadaqadaqaaiaad6gacqGHsislcaWGTbGaeyOeI0IaaG ymaiabgkHiTiabeQ7aRbGaayjkaiaawMcaaiaacgcaaaGaaiilaaaa @9646@ (35.1)

T 22 = B' 2 J nl1,2l+1 ( 2 ) = ( 2 s 2 +nm1 )! ( nm1 )! k=0 n (1)k Γ( nm+κ3 ) Γ( κ+2 ) ( 2 s 2 +k2 )! ( 2 s 2 +k )! 1 κ!( nm1κ )! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaikdacqGHsislcaaIYaaabeaajuaGcqGH9aqpcqGH sisldaWcaaqaaiaadkeacaGGNaaabaGaaGOmaaaadaqfWaqcfasabe aacaWGUbGaeyOeI0IaamiBaiabgkHiTiaaigdacaGGSaGaaGOmaiaa dYgacqGHRaWkcaaIXaaabaqcfa4aaeWaaKqbGeaacqGHsislcaaIYa aacaGLOaGaayzkaaaajuaGbaGaamOsaaaacqGH9aqpdaWcaaqaamaa bmaabaGaaGOmaiaadohadaWgaaqcfasaaiaaikdaaKqbagqaaiabgU caRiaad6gacqGHsislcaWGTbGaeyOeI0IaaGymaaGaayjkaiaawMca aiaacgcaaeaadaqadaqaaiaad6gacqGHsislcaWGTbGaeyOeI0IaaG ymaaGaayjkaiaawMcaaiaacgcaaaWaaabCaeaacaGGOaGaeyOeI0Ia aGymaiaacMcacaWGRbWaaSaaaeaacqqHtoWrdaqadaqaaiaad6gacq GHsislcaWGTbGaey4kaSIaeqOUdSMaeyOeI0IaaG4maaGaayjkaiaa wMcaaaqaaiabfo5ahnaabmaabaGaeyOeI0IaeqOUdSMaey4kaSIaaG OmaaGaayjkaiaawMcaaaaaaKqbGeaacaWGRbGaeyypa0JaaGimaaqa aiaad6gaaKqbakabggHiLdWaaSaaaeaadaqadaqaaiaaikdacaWGZb WaaSbaaKqbGeaacaaIYaaabeaajuaGcqGHRaWkcaWGRbGaeyOeI0Ia aGOmaaGaayjkaiaawMcaaiaacgcaaeaadaqadaqaaiaaikdacaWGZb WaaSbaaKqbGeaacaaIYaaajuaGbeaacqGHRaWkcaWGRbaacaGLOaGa ayzkaaGaaiyiaaaadaWcaaqaaiaaigdaaeaacqaH6oWAcaGGHaWaae WaaeaacaWGUbGaeyOeI0IaamyBaiabgkHiTiaaigdacqGHsislcqaH 6oWAaiaawIcacaGLPaaacaGGHaaaaaaa@9808@ (35.2)

T 32 = J nl1,2l+1 ( +1 ) = ( 2 s 2 +nm1 )! ( nm1 )! k=0 n (1)k Γ( nm+κ ) Γ( κ1 ) ( 2 s 2 +k+1 )! ( 2 s 2 +k )! 1 κ!( nm1κ )! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaiodacqGHsislcaaIYaaajuaGbeaacqGH9aqpdaqf WaqabKqbGeaacaWGUbGaeyOeI0IaamiBaiabgkHiTiaaigdacaGGSa GaaGOmaiaadYgacqGHRaWkcaaIXaaabaqcfa4aaeWaaKqbGeaacqGH RaWkcaaIXaaacaGLOaGaayzkaaaajuaGbaGaamOsaaaacqGH9aqpda WcaaqaamaabmaabaGaaGOmaiaadohadaWgaaqcfasaaiaaikdaaKqb agqaaiabgUcaRiaad6gacqGHsislcaWGTbGaeyOeI0IaaGymaaGaay jkaiaawMcaaiaacgcaaeaadaqadaqaaiaad6gacqGHsislcaWGTbGa eyOeI0IaaGymaaGaayjkaiaawMcaaiaacgcaaaWaaabCaeaacaGGOa GaeyOeI0IaaGymaiaacMcacaWGRbWaaSaaaeaacqqHtoWrdaqadaqa aiaad6gacqGHsislcaWGTbGaey4kaSIaeqOUdSgacaGLOaGaayzkaa aabaGaeu4KdC0aaeWaaeaacqGHsislcqaH6oWAcqGHsislcaaIXaaa caGLOaGaayzkaaaaaaqcfasaaiaadUgacqGH9aqpcaaIWaaabaGaam OBaaqcfaOaeyyeIuoadaWcaaqaamaabmaabaGaaGOmaiaadohadaWg aaqcfasaaiaaikdaaeqaaKqbakabgUcaRiaadUgacqGHRaWkcaaIXa aacaGLOaGaayzkaaGaaiyiaaqaamaabmaabaGaaGOmaiaadohadaWg aaqcfasaaiaaikdaaeqaaKqbakabgUcaRiaadUgaaiaawIcacaGLPa aacaGGHaaaamaalaaabaGaaGymaaqaaiabeQ7aRjaacgcadaqadaqa aiaad6gacqGHsislcaWGTbGaeyOeI0IaaGymaiabgkHiTiabeQ7aRb GaayjkaiaawMcaaiaacgcaaaaaaa@9326@ (35.3)

For (NC: 3D-RSP) symmetries, we have:

T 13 =A' J nl1,2l+1 ( 3 ) = ( 2l+1+nl1 )! ( nl1 )! k=0 n (1)k Γ( nl1+κ3 ) Γ( κ+3 ) ( 2l+1+k3 )! ( 2l+1+k )! 1 κ!( nl1κ )! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaigdacqGHsislcaaIZaaajuaGbeaacqGH9aqpcaWG bbGaai4jamaavadabeqcfasaaiaad6gacqGHsislcaWGSbGaeyOeI0 IaaGymaiaacYcacaaIYaGaamiBaiabgUcaRiaaigdaaeaajuaGdaqa daqcfasaaiabgkHiTiaaiodaaiaawIcacaGLPaaaaKqbagaacaWGkb aaaiabg2da9maalaaabaWaaeWaaeaacaaIYaGaamiBaiabgUcaRiaa igdacqGHRaWkcaWGUbGaeyOeI0IaamiBaiabgkHiTiaaigdaaiaawI cacaGLPaaacaGGHaaabaWaaeWaaeaacaWGUbGaeyOeI0IaamiBaiab gkHiTiaaigdaaiaawIcacaGLPaaacaGGHaaaamaaqahabaGaaiikai abgkHiTiaaigdacaGGPaGaam4AamaalaaabaGaeu4KdC0aaeWaaeaa caWGUbGaeyOeI0IaamiBaiabgkHiTiaaigdacqGHRaWkcqaH6oWAcq GHsislcaaIZaaacaGLOaGaayzkaaaabaGaeu4KdC0aaeWaaeaacqGH sislcqaH6oWAcqGHRaWkcaaIZaaacaGLOaGaayzkaaaaaaqcfasaai aadUgacqGH9aqpcaaIWaaabaGaamOBaaqcfaOaeyyeIuoadaWcaaqa amaabmaabaGaaGOmaiaadYgacqGHRaWkcaaIXaGaey4kaSIaam4Aai abgkHiTiaaiodaaiaawIcacaGLPaaacaGGHaaabaWaaeWaaeaacaaI YaGaamiBaiabgUcaRiaaigdacqGHRaWkcaWGRbaacaGLOaGaayzkaa GaaiyiaaaadaWcaaqaaiaaigdaaeaacqaH6oWAcaGGHaWaaeWaaeaa caWGUbGaeyOeI0IaamiBaiabgkHiTiaaigdacqGHsislcqaH6oWAai aawIcacaGLPaaacaGGHaaaaaaa@97EC@ (36.1)

T 31 = B' 2 J nl1,2l+1 ( 2 ) = ( 2l+1+nl1 )! ( nl1 )! k=0 n (1)k Γ( nl1+κ2 ) Γ( κ+2 ) ( 2l+1+k2 )! ( 2l+1+k )! 1 κ!( nl1κ )! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiabgkHiTiaaiodacaaIXaaajuaGbeaacqGH9aqpcqGH sisldaWcaaqaaiaadkeacaGGNaaabaGaaGOmaaaadaqfWaqabKqbGe aacaWGUbGaeyOeI0IaamiBaiabgkHiTiaaigdacaGGSaGaaGOmaiaa dYgacqGHRaWkcaaIXaaabaqcfa4aaeWaaKqbGeaacqGHsislcaaIYa aacaGLOaGaayzkaaaajuaGbaGaamOsaaaacqGH9aqpdaWcaaqaamaa bmaabaGaaGOmaiaadYgacqGHRaWkcaaIXaGaey4kaSIaamOBaiabgk HiTiaadYgacqGHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaqaamaa bmaabaGaamOBaiabgkHiTiaadYgacqGHsislcaaIXaaacaGLOaGaay zkaaGaaiyiaaaadaaeWbqaaiaacIcacqGHsislcaaIXaGaaiykaiaa dUgadaWcaaqaaiabfo5ahnaabmaabaGaamOBaiabgkHiTiaadYgacq GHsislcaaIXaGaey4kaSIaeqOUdSMaeyOeI0IaaGOmaaGaayjkaiaa wMcaaaqaaiabfo5ahnaabmaabaGaeyOeI0IaeqOUdSMaey4kaSIaaG OmaaGaayjkaiaawMcaaaaaaKqbGeaacaWGRbGaeyypa0JaaGimaaqa aiaad6gaaKqbakabggHiLdWaaSaaaeaadaqadaqaaiaaikdacaWGSb Gaey4kaSIaaGymaiabgUcaRiaadUgacqGHsislcaaIYaaacaGLOaGa ayzkaaGaaiyiaaqaamaabmaabaGaaGOmaiaadYgacqGHRaWkcaaIXa Gaey4kaSIaam4AaaGaayjkaiaawMcaaiaacgcaaaWaaSaaaeaacaaI XaaabaGaeqOUdSMaaiyiamaabmaabaGaamOBaiabgkHiTiaadYgacq GHsislcaaIXaGaeyOeI0IaeqOUdSgacaGLOaGaayzkaaGaaiyiaaaa aaa@99A2@ (36.2)

T 13 = J nl1,2l+1 ( +1 ) = ( 2l+1+nl1 )! ( nl1 )! k=0 n (1)k Γ( nl1+κ+1 ) Γ( κ1 ) ( 2l+1+k+1 )! ( 2l+1+k )! 1 κ!( nl1κ )! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaigdacqGHsislcaaIZaaajuaGbeaacqGH9aqpdaqf WaqabKqbGeaacaWGUbGaeyOeI0IaamiBaiabgkHiTiaaigdacaGGSa GaaGOmaiaadYgacqGHRaWkcaaIXaaabaqcfa4aaeWaaKqbGeaacqGH RaWkcaaIXaaacaGLOaGaayzkaaaajuaGbaGaamOsaaaacqGH9aqpda WcaaqaamaabmaabaGaaGOmaiaadYgacqGHRaWkcaaIXaGaey4kaSIa amOBaiabgkHiTiaadYgacqGHsislcaaIXaaacaGLOaGaayzkaaGaai yiaaqaamaabmaabaGaamOBaiabgkHiTiaadYgacqGHsislcaaIXaaa caGLOaGaayzkaaGaaiyiaaaadaaeWbqaaiaacIcacqGHsislcaaIXa GaaiykaiaadUgadaWcaaqaaiabfo5ahnaabmaabaGaamOBaiabgkHi TiaadYgacqGHsislcaaIXaGaey4kaSIaeqOUdSMaey4kaSIaaGymaa GaayjkaiaawMcaaaqaaiabfo5ahnaabmaabaGaeyOeI0IaeqOUdSMa eyOeI0IaaGymaaGaayjkaiaawMcaaaaaaKqbGeaacaWGRbGaeyypa0 JaaGimaaqaaiaad6gaaKqbakabggHiLdWaaSaaaeaadaqadaqaaiaa ikdacaWGSbGaey4kaSIaaGymaiabgUcaRiaadUgacqGHRaWkcaaIXa aacaGLOaGaayzkaaGaaiyiaaqaamaabmaabaGaaGOmaiaadYgacqGH RaWkcaaIXaGaey4kaSIaam4AaaGaayjkaiaawMcaaiaacgcaaaWaaS aaaeaacaaIXaaabaGaeqOUdSMaaiyiamaabmaabaGaamOBaiabgkHi TiaadYgacqGHsislcaaIXaGaeyOeI0IaeqOUdSgacaGLOaGaayzkaa Gaaiyiaaaaaaa@965D@ (36.3)

Which allow us to obtaining the exact modifications of fundamental states ( E ncper:u ( θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeqiUdeNaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@4533@  , E ncper:D ( θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadseaaKqbagqaamaabmaabaGaeqiUdeNaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@4502@ ) and ( E ncper:u ( Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@44F4@ , E ncper:D ( Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadseaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@44C3@ ) produced by spin-orbital effect:

E ncper:u ( θ, θ ¯ ) 2 p + ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( θ T s2is ( A,B,n,l )+ θ ¯ 2 m 0 T 32 ) E ncper:D ( θ, θ ¯ ) 2 p + ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( θ T s2is ( A,B,n,l )+ θ ¯ 2 m 0 T 32 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGWbGaamyzaiaa dkhacaGG6aGaamyDaaqabaqcfa4aaeWaaeaacqaH4oqCcaGGSaGafq iUdeNbaebaaiaawIcacaGLPaaacqGHHjIUdaWcaaqaaiaaikdacqGH pis1caWGWbWaaSbaaeaacqGHRaWkaeqaaaqaamaabmaabaGaeyOeI0 IaaGioaiaadweaaiaawIcacaGLPaaaaaWaaeWaaeaadaWcaaqaaiaa isdacqqHsoGqaeaacaaIYaGaamOBaiabgkHiTiaaikdacaWGTbGaey 4kaSIaaGOmaiaadohadaWgaaqcfasaaiaaikdaaKqbagqaaiabgkHi TiaaigdaaaaacaGLOaGaayzkaaWaaWbaaKqbGeqabaGaaGOmaaaaju aGdaqadaqaamaalaaabaWaaeWaaeaacaWGUbGaeyOeI0IaamyBaiab gkHiTiaaigdaaiaawIcacaGLPaaacaGGHaaabaWaaeWaaeaacaaIYa GaamOBaiabgkHiTiaaikdacaWGTbGaey4kaSIaaGOmaiaadohadaWg aaqcfasaaiaaikdaaeqaaKqbakabgkHiTiaaigdaaiaawIcacaGLPa aadaqadaqaaiaad6gacqGHsislcaWGTbGaey4kaSIaaGOmaiaadoha daWgaaqcfasaaiaaikdaaeqaaKqbakabgkHiTiaaigdaaiaawIcaca GLPaaacaGGHaaaaaGaayjkaiaawMcaamaabmaabaGaeqiUdeNaamiv amaaBaaabaGaam4CaKqbGiaaikdajuaGcqGHsislcaWGPbGaam4Caa qabaWaaeWaaeaacaWGbbGaaiilaiaadkeacaGGSaGaamOBaiaacYca caWGSbaacaGLOaGaayzkaaGaey4kaSYaaSaaaeaacuaH4oqCgaqeaa qaaiaaikdadaqfqaqabKqbGeaacaaIWaaajuaGbeqaaiaad2gaaaaa aiaadsfadaWgaaqcfasaaiaaiodacqGHsislcaaIYaaajuaGbeaaai aawIcacaGLPaaaaOqaaKqbakaadweadaWgaaqcfasaaiaad6gacaWG JbGaeyOeI0IaamiCaiaadwgacaWGYbGaaiOoaiaadseaaKqbagqaam aabmaabaGaeqiUdeNaaiilaiqbeI7aXzaaraaacaGLOaGaayzkaaGa eyyyIO7aaSaaaeaacaaIYaGaey4dIuTaamiCamaaBaaabaGaey4kaS cabeaaaeaadaqadaqaaiabgkHiTiaaiIdacaWGfbaacaGLOaGaayzk aaaaamaabmaabaWaaSaaaeaacaaI0aGaeuOKdieabaGaaGOmaiaad6 gacqGHsislcaaIYaGaamyBaiabgUcaRiaaikdacaWGZbWaaSbaaKqb GeaacaaIYaaajuaGbeaacqGHsislcaaIXaaaaaGaayjkaiaawMcaam aaCaaajuaibeqaaiaaikdaaaqcfa4aaeWaaeaadaWcaaqaamaabmaa baGaamOBaiabgkHiTiaad2gacqGHsislcaaIXaaacaGLOaGaayzkaa GaaiyiaaqaamaabmaabaGaaGOmaiaad6gacqGHsislcaaIYaGaamyB aiabgUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaIYaaabeaajuaGcq GHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacaWGUbGaeyOeI0Ia amyBaiabgUcaRiaaikdacaWGZbWaaSbaaKazfa4=baGaaGOmaaqcfa yabaGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaacgcaaaaacaGLOaGa ayzkaaWaaeWaaeaacqaH4oqCcaWGubWaaSbaaeaacaWGZbqcfaIaaG OmaKqbakabgkHiTiaadMgacaWGZbaabeaadaqadaqaaiaadgeacaGG SaGaamOqaiaacYcacaWGUbGaaiilaiaadYgaaiaawIcacaGLPaaacq GHRaWkdaWcaaqaaiqbeI7aXzaaraaabaGaaGOmamaavababeqcfasa aiaaicdaaKqbagqabaGaamyBaaaaaaGaamivamaaBaaajuaibaGaaG 4maiabgkHiTiaaikdaaeqaaKqbaoaaBaaajuaibaGaaG4maaqabaaa juaGcaGLOaGaayzkaaaaaaa@FDCE@ (37.1)

and
E ncper:u ( Θ, θ ¯ ) α p + ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ( Θ T s3is ( A,B,n,l )+ θ ¯ 2 m 0 T 33 ) E ncper:D ( Θ, θ ¯ ) α p ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ( Θ T s3is ( A,B,n,l )+ θ ¯ 2 m 0 T 33 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGWbGaamyzaiaa dkhacaGG6aGaamyDaaqcfayabaWaaeWaaeaacqqHyoqucaGGSaGafq iUdeNbaebaaiaawIcacaGLPaaacqGHHjIUcqGHsisldaWcaaqaaiab eg7aHjaadchadaWgaaqaaiabgUcaRaqabaaabaWaaeWaaeaacqGHsi slcaaI4aGaamyraaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaioda caGGVaGaaGOmaaaaaaqcfa4aaeWaaeaadaWcaaqaaiaaikdacaWGcb aabaGaamOBaaaaaiaawIcacaGLPaaadaahaaqcfasabeaacaaIZaaa aKqbaoaalaaabaWaaeWaaeaacaWGUbGaeyOeI0IaamiBaiabgkHiTi aaigdaaiaawIcacaGLPaaacaGGHaaabaGaaGOmaiaad6gadaqadaqa aiaad6gacqGHRaWkcaWGSbaacaGLOaGaayzkaaGaaiyiaaaadaqada qaaiabfI5arjaadsfadaWgaaqcfasaaiaadohacaaIZaGaeyOeI0Ia amyAaiaadohaaeqaaKqbaoaabmaabaGaamyqaiaacYcacaWGcbGaai ilaiaad6gacaGGSaGaamiBaaGaayjkaiaawMcaaiabgUcaRmaalaaa baGafqiUdeNbaebaaeaacaaIYaWaaubeaeqajuaibaGaaGimaaqcfa yabeaacaWGTbaaaaaacaWGubWaaSbaaKqbGeaacaaIZaGaeyOeI0Ia aG4maaqabaaajuaGcaGLOaGaayzkaaaakeaajuaGcaWGfbWaaSbaaK qbGeaacaWGUbGaam4yaiabgkHiTiaadchacaWGLbGaamOCaiaacQda caWGebaajuaGbeaadaqadaqaaiabfI5arjaacYcacuaH4oqCgaqeaa GaayjkaiaawMcaaiabggMi6kabgkHiTmaalaaabaGaeqySdeMaamiC amaaBaaabaGaeyOeI0cabeaaaeaadaqadaqaaiabgkHiTiaaiIdaca WGfbaacaGLOaGaayzkaaWaaWbaaKqbGeqabaGaaG4maiaac+cacaaI YaaaaaaajuaGdaqadaqaamaalaaabaGaaGOmaiaadkeaaeaacaWGUb aaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaiodaaaqcfa4aaSaa aeaadaqadaqaaiaad6gacqGHsislcaWGSbGaeyOeI0IaaGymaaGaay jkaiaawMcaaiaacgcaaeaacaaIYaGaamOBamaabmaabaGaamOBaiab gUcaRiaadYgaaiaawIcacaGLPaaacaGGHaaaamaabmaabaGaeuiMde LaamivamaaBaaajuaibaGaam4CaiaaiodacqGHsislcaWGPbGaam4C aaqabaqcfa4aaeWaaeaacaWGbbGaaiilaiaadkeacaGGSaGaamOBai aacYcacaWGSbaacaGLOaGaayzkaaGaey4kaSYaaSaaaeaacuaH4oqC gaqeaaqaaiaaikdadaqfqaqabKqbGeaacaaIWaaajuaGbeqaaiaad2 gaaaaaaiaadsfadaWgaaqcfasaaiaaiodacqGHsislcaaIZaaabeaa aKqbakaawIcacaGLPaaaaaaa@C9B0@ (37.2)
Where, the two factors T s2is ( A,B,n,l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaadohacaaIYaGaeyOeI0IaamyAaiaadohaaeqaaKqb aoaabmaabaGaamyqaiaacYcacaWGcbGaaiilaiaad6gacaGGSaGaam iBaaGaayjkaiaawMcaaaaa@43C0@  and T s3is ( A,B,n,l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaadohacaaIZaGaeyOeI0IaamyAaiaadohaaeqaaKqb aoaabmaabaGaamyqaiaacYcacaWGcbGaaiilaiaad6gacaGGSaGaam iBaaGaayjkaiaawMcaaaaa@43C1@  are given by, respectively:
T s2is ( A,B,n,l )= i=1 2 T i2 T s3is ( A,B,n,l )= i=1 2 T i3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam ivamaaBaaajuaibaGaam4CaiaaikdacqGHsislcaWGPbGaam4Caaqa baqcfa4aaeWaaeaacaWGbbGaaiilaiaadkeacaGGSaGaamOBaiaacY cacaWGSbaacaGLOaGaayzkaaGaeyypa0ZaaabCaeaacaWGubWaaSba aKqbGeaacaWGPbGaeyOeI0IaaGOmaaqcfayabaaajuaqbaGaamyAai abg2da9iaaigdaaKqbGeaacaaIYaaajuaGcqGHris5aaGcbaqcfaOa amivamaaBaaajuaibaGaam4CaiaaiodacqGHsislcaWGPbGaam4Caa qabaqcfa4aaeWaaeaacaWGbbGaaiilaiaadkeacaGGSaGaamOBaiaa cYcacaWGSbaacaGLOaGaayzkaaGaeyypa0ZaaabCaeaacaWGubWaaS baaKqbGeaacaWGPbGaeyOeI0IaaG4maaqcfayabaaajuaibaGaamyA aiabg2da9iaaigdaaeaacaaIYaaajuaGcqGHris5aaaaaa@693E@ (38)

The exact magnetic spectrum for (m.i.s.) potential in both (NC: 2D- RSP) and (NC: 3D- RSP) symmetries for excited states for one-electron atoms

 Having obtained the exact modifications to the energy levels ( E ncper:u ( θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeqiUdeNaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@4533@ , E ncper:D ( θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadseaaKqbagqaamaabmaabaGaeqiUdeNaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@4502@ ) and ( E ncper:u ( Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@44F4@ , E ncper:D ( Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadseaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiqbeI7aXz aaraaacaGLOaGaayzkaaaaaa@44C3@ ), for exited n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada ahaaqcfasabeaacaWG0bGaamiAaaaaaaa@39A2@  states, produced with spin-orbital induced Hamiltonians operators, we now consider interested physically meaningful phenomena, which produced from the perturbative terms of inverse-square potential related to the influence of an external uniform magnetic field, it’s sufficient to apply the following three replacements to describing these phenomena:
L z θ ¯ 2 m 0 +( A r 4 B 2 r 3 )θ L z { σ ¯ 2 m 0 +χ( A r 4 B 2 r 3 ) } H L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaamitamaaBaaajuaibaGaamOEaaqabaqcfa4aa0aaaeaacqaH4oqC aaaabaGaaGOmaiaad2gadaWgaaqcfasaaiaaicdaaeqaaaaajuaGcq GHRaWkdaqadaqaamaalaaabaGaamyqaaqaaiaadkhadaahaaqabKqb GeaacaaI0aaaaaaajuaGcqGHsisldaWcaaqaaiaadkeaaeaacaaIYa GaamOCamaaCaaajuaibeqaaiaaiodaaaaaaaqcfaOaayjkaiaawMca aiabeI7aXjaadYeadaWgaaqcfasaaiaadQhaaKqbagqaaiabgkziUo aacmaabaWaaSaaaeaadaqdaaqaaiabeo8aZbaaaeaacaaIYaGaamyB amaaBaaajuaibaGaaGimaaqcfayabaaaaiabgUcaRiabeE8aJnaabm aabaWaaSaaaeaacaWGbbaabaGaamOCamaaCaaabeqcfasaaiaaisda aaaaaKqbakabgkHiTmaalaaabaGaamOqaaqaaiaaikdacaWGYbWaaW baaKqbGeqabaGaaG4maaaaaaaajuaGcaGLOaGaayzkaaaacaGL7bGa ayzFaaGaaGPaVpaaFiaabaGaamisaaGaay51GaWaa8HaaeaacaWGmb aacaGLxdcaaaa@6A00@ (39.1)
L θ ¯ 2 m 0 +( A r 4 B 2 r 3 ) L Θ { σ ¯ 2 m 0 +χ( A r 4 B 2 r 3 ) } H L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba acbeGab8htayaalaWaa8HaaeaadaqdaaqaaGGabiab+H7aXbaaaiaa wEniaaqaaiaaikdacaWGTbWaaSbaaKqbGeaacaaIWaaajuaGbeaaaa Gaey4kaSYaaeWaaeaadaWcaaqaaiaadgeaaeaacaWGYbWaaWbaaeqa juaibaGaaGinaaaaaaqcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaaG OmaiaadkhadaahaaqcfasabeaacaaIZaaaaaaaaKqbakaawIcacaGL PaaaceWFmbGbaSaacuqHyoqugaWcaiabgkziUoaacmaabaWaaSaaae aadaqdaaqaaiabeo8aZbaaaeaacaaIYaGaamyBamaaBaaajuaibaGa aGimaaqabaaaaKqbakabgUcaRiabeE8aJnaabmaabaWaaSaaaeaaca WGbbaabaGaamOCamaaCaaabeqcfasaaiaaisdaaaaaaKqbakabgkHi TmaalaaabaGaamOqaaqaaiaaikdacaWGYbWaaWbaaKqbGeqabaGaaG 4maaaaaaaajuaGcaGLOaGaayzkaaaacaGL7bGaayzFaaWaa8Haaeaa caWGibaacaGLxdcadaWhcaqaaiaadYeaaiaawEniaaaa@666F@ (39.2)
θχH,ΘχH    and  θ ¯ σ ¯ H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXj abgkziUkabeE8aJLqbGiaadIeajuaGcaGGSaGaeuiMdeLaeyOKH4Qa eq4XdmwcfaIaamisaKqbakaabccacaqGGaGaaeiiaiaabccacaqGHb GaaeOBaiaabsgacaqGGaGafqiUdeNbaebacqGHsgIRdaqdaaqaaiab eo8aZbaacaWGibaaaa@50F7@ (39.3)
Here and  are infinitesimal real proportional’s constants, and we choose the magnetic field  H =H k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFmaaba GaaeiiaiaadIeaaiaawgoiaiabg2da9iaadIeadaWhdaqaaiaadUga aiaawgoiaaaa@3E2A@ , which allow us to introduce the modified new magnetic Hamiltonians H ^ m2is ( r,A,B,χ, σ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaaikdacqGHsislcaWGPbGaam4Caaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiaadgeacaGGSaGaamOqaiaacY cacqaHhpWycaGGSaWaa0aaaeaacqaHdpWCaaaacaGLOaGaayzkaaaa aa@470C@  and H ^ m3is ( r,A,B,χ, σ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaaiodacqGHsislcaWGPbGaam4Caaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiaadgeacaGGSaGaamOqaiaacY cacqaHhpWycaGGSaWaa0aaaeaacqaHdpWCaaaacaGLOaGaayzkaaaa aa@470D@  in (NC: 2D-RSP) and (NC: 3D-RSP), respectively, as:
H ^ m2is ( r,A,B,χ, σ ¯ )=( χ( A r 4 B 2 r 3 )+ σ ¯ 2 m 0 )( H J S H ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaaikdacqGHsislcaWGPbGaam4Caaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiaadgeacaGGSaGaamOqaiaacY cacqaHhpWycaGGSaWaa0aaaeaacqaHdpWCaaaacaGLOaGaayzkaaGa eyypa0ZaaeWaaeaacqaHhpWydaqadaqaamaalaaabaGaamyqaaqaai aadkhadaahaaqcfasabeaacaaI0aaaaaaajuaGcqGHsisldaWcaaqa aiaadkeaaeaacaaIYaGaamOCamaaCaaabeqcfasaaiaaiodaaaaaaa qcfaOaayjkaiaawMcaaiaaykW7cqGHRaWkdaWcaaqaamaanaaabaGa eq4WdmhaaaqaaiaaikdacaWGTbWaaSbaaKqbGeaacaaIWaaajuaGbe aaaaaacaGLOaGaayzkaaWaaeWaaeaadaWhcaqaaiaadIeaaiaawEni amaaFiaabaGaamOsaaGaay51GaGaeyOeI0Yaa8XaaeaacaWGtbaaca GLHdcadaWhdaqaaiaadIeaaiaawgoiaaGaayjkaiaawMcaaaaa@6989@ (40.1)
and
H ^ m3is ( r,A,B,χ, σ ¯ )=( χ( A r 4 B 2 r 3 )+ σ ¯ 2 m 0 )( H J S H ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaaiodacqGHsislcaWGPbGaam4Caaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiaadgeacaGGSaGaamOqaiaacY cacqaHhpWycaGGSaWaa0aaaeaacqaHdpWCaaaacaGLOaGaayzkaaGa eyypa0ZaaeWaaeaacqaHhpWydaqadaqaamaalaaabaGaamyqaaqaai aadkhadaahaaqabKqbGeaacaaI0aaaaaaajuaGcqGHsisldaWcaaqa aiaadkeaaeaacaaIYaGaamOCamaaCaaabeqcfasaaiaaiodaaaaaaa qcfaOaayjkaiaawMcaaiaaykW7cqGHRaWkdaWcaaqaamaanaaabaGa eq4WdmhaaaqaaiaaikdacaWGTbWaaSbaaKqbGeaacaaIWaaajuaGbe aaaaaacaGLOaGaayzkaaWaaeWaaeaadaWhcaqaaiaadIeaaiaawEni amaaFiaabaGaamOsaaGaay51GaGaeyOeI0Yaa8XaaeaacaWGtbaaca GLHdcadaWhdaqaaiaadIeaaiaawgoiaaGaayjkaiaawMcaaaaa@698A@ (40.2)
Here ( S H ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeyOeI0Yaa8XaaeaacaWGtbaacaGLHdcadaWhdaqaaiaadIeaaiaa wgoiaaGaayjkaiaawMcaaaaa@3E12@  denote to the ordinary Hamiltonian of Zeeman Effect. To obtain the exact noncommutative magnetic modifications of energy ( E mag2-is ( θ, θ ¯ ,n,m,A,B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaabkdacaqGTaGaaeyAaiaa bohaaKqbagqaamaabmaabaGaeqiUdeNaaiilaiqbeI7aXzaaraGaai ilaiaad6gacaGGSaGaamyBaiaacYcacaWGbbGaaiilaiaadkeaaiaa wIcacaGLPaaaaaa@4A14@ , E mag-3is ( Θ, θ ¯ ,n,l,A,B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaab2cacaqGZaGaaeyAaiaa bohaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiqbeI7aXzaaraGaai ilaiaad6gacaGGSaGaamiBaiaacYcacaWGbbGaaiilaiaadkeaaiaa wIcacaGLPaaaaaa@49D5@ ) for modified inverse-square potential, which produced automatically by the effect of H ^ m2is ( r,A,B,χ, σ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaaikdacqGHsislcaWGPbGaam4Caaqc fayabaWaaeWaaeaacaWGYbGaaiilaiaadgeacaGGSaGaamOqaiaacY cacqaHhpWycaGGSaWaa0aaaeaacqaHdpWCaaaacaGLOaGaayzkaaaa aa@470C@  and H ^ m3is ( r,A,B,χ, σ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaaiodacqGHsislcaWGPbGaam4Caaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiaadgeacaGGSaGaamOqaiaacY cacqaHhpWycaGGSaWaa0aaaeaacqaHdpWCaaaacaGLOaGaayzkaaaa aa@470D@ , we make the following three simultaneously replacements:
p + m,( θ,Θ )( χ,χ )    and      θ ¯ σ ¯ H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadchada WgaaqaaiabgUcaRaqabaGaeyOKH4QaaeyBaiaabYcadaqadaqaaiab eI7aXjaacYcacqqHyoquaiaawIcacaGLPaaacqGHsgIRdaqadaqaai abeE8aJjaacYcacqaHhpWyaiaawIcacaGLPaaacaqGGaGaaeiiaiaa bccacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaabccacaqGGaGaae iiaiaabccadaqdaaqaaiabeI7aXbaacqGHsgIRdaqdaaqaaiabeo8a ZbaacaWGibaaaa@57C3@ (41)
In two Eqs. (37.1) and (37.2) to obtain the two values E mag2-is ( θ, θ ¯ ,n,m,A,B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaabkdacaqGTaGaaeyAaiaa bohaaeqaaKqbaoaabmaabaGaeqiUdeNaaiilaiqbeI7aXzaaraGaai ilaiaad6gacaGGSaGaamyBaiaacYcacaWGbbGaaiilaiaadkeaaiaa wIcacaGLPaaaaaa@4A14@  and E mag-3is ( Θ, θ ¯ ,n,l,A,B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaab2cacaqGZaGaaeyAaiaa bohaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiqbeI7aXzaaraGaai ilaiaad6gacaGGSaGaamiBaiaacYcacaWGbbGaaiilaiaadkeaaiaa wIcacaGLPaaaaaa@49D5@ for the exact magnetic modifications of spectrum corresponding n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada ahaaqcfasabeaacaWG0bGaamiAaaaaaaa@39A2@  excited states, in (NC-2D: RSP) and (NC-3D: RSP), respectively, as:

E mag2-is ( θ, θ ¯ ,n,m,A,B ) 2mH ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( χ T s2is ( A,B,n,l )+ σ ¯ 2 m 0 T 32 ( A,B,n,l ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaabkdacaqGTaGaaeyAaiaa bohaaKqbagqaamaabmaabaGaeqiUdeNaaiilaiqbeI7aXzaaraGaai ilaiaad6gacaGGSaGaamyBaiaacYcacaWGbbGaaiilaiaadkeaaiaa wIcacaGLPaaacqGHHjIUdaWcaaqaaiaaikdacqGHpis1caWGTbGaam isaaqaamaabmaabaGaeyOeI0IaaGioaiaadweaaiaawIcacaGLPaaa aaWaaeWaaeaadaWcaaqaaiaaisdacqqHsoGqaeaacaaIYaGaamOBai abgkHiTiaaikdacaWGTbGaey4kaSIaaGOmaiaadohadaWgaaqcfasa aiaaikdaaKqbagqaaiabgkHiTiaaigdaaaaacaGLOaGaayzkaaWaaW baaKqbGeqabaGaaGOmaaaajuaGdaqadaqaamaalaaabaWaaeWaaeaa caWGUbGaeyOeI0IaamyBaiabgkHiTiaaigdaaiaawIcacaGLPaaaca GGHaaabaWaaeWaaeaacaaIYaGaamOBaiabgkHiTiaaikdacaWGTbGa ey4kaSIaaGOmaiaadohadaWgaaqcfasaaiaaikdaaeqaaKqbakabgk HiTiaaigdaaiaawIcacaGLPaaadaqadaqaaiaad6gacqGHsislcaWG TbGaey4kaSIaaGOmaiaadohadaWgaaqcfasaaiaaikdaaeqaaKqbak abgkHiTiaaigdaaiaawIcacaGLPaaacaGGHaaaaaGaayjkaiaawMca amaabmaabaGaeq4XdmMaamivamaaBaaajuaibaGaam4Caiaaikdacq GHsislcaWGPbGaam4Caaqabaqcfa4aaeWaaeaacaWGbbGaaiilaiaa dkeacaGGSaGaamOBaiaacYcacaWGSbaacaGLOaGaayzkaaGaey4kaS YaaSaaaeaadaqdaaqaaiabeo8aZbaaaeaacaaIYaWaaubeaeqajuai baGaaGimaaqcfayabeaacaWGTbaaaaaacaWGubWaaSbaaKqbGeaaca aIZaGaeyOeI0IaaGOmaaqabaqcfa4aaeWaaeaacaWGbbGaaiilaiaa dkeacaGGSaGaamOBaiaacYcacaWGSbaacaGLOaGaayzkaaaacaGLOa GaayzkaaGaaGPaVdaa@A582@  (42.1)

and
E mag-3is ( Θ, θ ¯ ,n,l,A,B )= mH ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ( χ T s3is ( A,B,n,l )+ σ ¯ 2 m 0 T 33 ( A,B,n,l ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaab2cacaqGZaGaaeyAaiaa bohaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiqbeI7aXzaaraGaai ilaiaad6gacaGGSaGaamiBaiaacYcacaWGbbGaaiilaiaadkeaaiaa wIcacaGLPaaacqGH9aqpcqGHsisldaWcaaqaaiaad2gacaWGibaaba WaaeWaaeaacqGHsislcaaI4aGaamyraaGaayjkaiaawMcaamaaCaaa beqcfasaaiaaiodacaGGVaGaaGOmaaaaaaqcfa4aaeWaaeaadaWcaa qaaiaaikdacaWGcbaabaGaamOBaaaaaiaawIcacaGLPaaadaahaaqc fasabeaacaaIZaaaaKqbaoaalaaabaWaaeWaaeaacaWGUbGaeyOeI0 IaamiBaiabgkHiTiaaigdaaiaawIcacaGLPaaacaGGHaaabaGaaGOm aiaad6gadaqadaqaaiaad6gacqGHRaWkcaWGSbaacaGLOaGaayzkaa GaaiyiaaaadaqadaqaaiabeE8aJjaadsfadaWgaaqcfasaaiaadoha caaIZaGaeyOeI0IaamyAaiaadohaaKqbagqaamaabmaabaGaamyqai aacYcacaWGcbGaaiilaiaad6gacaGGSaGaamiBaaGaayjkaiaawMca aiabgUcaRmaalaaabaWaa0aaaeaacqaHdpWCaaaabaGaaGOmamaava babeqcfasaaiaaicdaaKqbagqabaGaamyBaaaaaaGaamivamaaBaaa juaibaGaaG4maiabgkHiTiaaiodaaeqaaKqbaoaabmaabaGaamyqai aacYcacaWGcbGaaiilaiaad6gacaGGSaGaamiBaaGaayjkaiaawMca aaGaayjkaiaawMcaaiaaykW7aaa@8AF9@ (42.2)
Where denote to the angular momentum quantum number, lm+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadYgacqGHKjYOcaWGTbGaeyizImQaey4kaSIaamiBaaaa@3E86@ , which allow us to fixing ( 2l+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdaca WGSbGaey4kaSIaaGymaaaa@39C3@ ) values for the orbital angular momentum quantum numbers.

Results of Exact Modified Global Spectrum of the Lowest Excitations States for (m.i.s.) Potential in both (nc:2d- rsp) and (nc:3d- rsp) Symmetries for One-Electron Atoms

 Let us now resume the eigenenergies of the modified Schrödinger equations obtained in this paper, the total modified energies ( E ncu ( θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamyDaaqabaqcfa4aaeWa aeaacqaH4oqCcaGGSaGafqiUdeNbaebaaiaawIcacaGLPaaaaaa@419F@ , E ncD ( θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamiraaqabaqcfa4aaeWa aeaacqaH4oqCcaGGSaGafqiUdeNbaebaaiaawIcacaGLPaaaaaa@416E@ ) and ( E ncu ( Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamyDaaqabaqcfa4aaeWa aeaacqqHyoqucaGGSaGafqiUdeNbaebaaiaawIcacaGLPaaaaaa@4160@ , E ncD ( Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamiraaqabaqcfa4aaeWa aeaacqqHyoqucaGGSaGafqiUdeNbaebaaiaawIcacaGLPaaaaaa@412F@ ) of a particle fermionic with spin up and spin down are determined corresponding excited states, respectively, for modified inverse-square potential in (NC: 2D-RSP) and (NC: 3D-RSP), on based to the obtained new results (10.a), (37.1), (37.2), (41.1), (41.2) and (37.b), in addition to the original results (17) of energies we obtain the four new values of global energies:

E ncu ( θ, θ ¯ ) 2Β 2 ( 2n2m1+ 2A+ m 2 ) 2 + 2 p + ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( θ T sis ( A,B,n,l )+ θ ¯ 2 m 0 T 3 ) + 2mH ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( χ T s2is ( A,B,n,l )+ σ ¯ 2 m 0 T 32 ( A,B,n,l ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWG1baabeaajuaG daqadaqaaiabeI7aXjaacYcacuaH4oqCgaqeaaGaayjkaiaawMcaai abggMi6kabgkHiTmaalaaabaWaaubiaeqabeqcfasaaiaaikdaaKqb agaacaaIYaGaeuOKdieaaaqaamaavacabeqabKqbGeaacaaIYaaaju aGbaWaaeWaaeaacaaIYaGaamOBaiabgkHiTiaaikdacaWGTbGaeyOe I0IaaGymaiabgUcaRmaakaaabaGaaGOmaiaadgeacqGHRaWkcaWGTb WaaWbaaeqabaGaaGOmaaaaaeqaaaGaayjkaiaawMcaaaaaaaGaey4k aSYaaSaaaeaacaaIYaGaey4dIuTaamiCamaaBaaabaGaey4kaScabe aaaeaadaqadaqaaiabgkHiTiaaiIdacaWGfbaacaGLOaGaayzkaaaa amaabmaabaWaaSaaaeaacaaI0aGaeuOKdieabaGaaGOmaiaad6gacq GHsislcaaIYaGaamyBaiabgUcaRiaaikdacaWGZbWaaSbaaKqbGeaa caaIYaaabeaajuaGcqGHsislcaaIXaaaaaGaayjkaiaawMcaamaaCa aajuaibeqaaiaaikdaaaqcfa4aaeWaaeaadaWcaaqaamaabmaabaGa amOBaiabgkHiTiaad2gacqGHsislcaaIXaaacaGLOaGaayzkaaGaai yiaaqaamaabmaabaGaaGOmaiaad6gacqGHsislcaaIYaGaamyBaiab gUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaIYaaabeaajuaGcqGHsi slcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacaWGUbGaeyOeI0IaamyB aiabgUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaIYaaajuaGbeaacq GHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaaaaiaawIcacaGLPaaa daqadaqaaiabeI7aXjaadsfadaWgaaqcfasaaiaadohacqGHsislca WGPbGaam4CaaqcfayabaWaaeWaaeaacaWGbbGaaiilaiaadkeacaGG SaGaamOBaiaacYcacaWGSbaacaGLOaGaayzkaaGaey4kaSYaaSaaae aacuaH4oqCgaqeaaqaaiaaikdadaqfqaqabKqbGeaacaaIWaaajuaG beqaaiaad2gaaaaaaiaadsfadaWgaaqcfasaaiaaiodaaKqbagqaaa GaayjkaiaawMcaaaGcbaqcfaOaey4kaSYaaSaaaeaacaaIYaGaey4d IuTaamyBaiaadIeaaeaadaqadaqaaiabgkHiTiaaiIdacaWGfbaaca GLOaGaayzkaaaaamaabmaabaWaaSaaaeaacaaI0aGaeuOKdieabaGa aGOmaiaad6gacqGHsislcaaIYaGaamyBaiabgUcaRiaaikdacaWGZb WaaSbaaKqbGeaacaaIYaaajuaGbeaacqGHsislcaaIXaaaaaGaayjk aiaawMcaamaaCaaajuaibeqaaiaaikdaaaqcfa4aaeWaaeaadaWcaa qaamaabmaabaGaamOBaiabgkHiTiaad2gacqGHsislcaaIXaaacaGL OaGaayzkaaGaaiyiaaqaamaabmaabaGaaGOmaiaad6gacqGHsislca aIYaGaamyBaiabgUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaIYaaa beaajuaGcqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacaWGUb GaeyOeI0IaamyBaiabgUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaI YaaabeaajuaGcqGHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaaaai aawIcacaGLPaaadaqadaqaaiabeE8aJjaadsfadaWgaaqaaiaadoha juaicaaIYaqcfaOaeyOeI0IaamyAaiaadohaaeqaamaabmaabaGaam yqaiaacYcacaWGcbGaaiilaiaad6gacaGGSaGaamiBaaGaayjkaiaa wMcaaiabgUcaRmaalaaabaWaa0aaaeaacqaHdpWCaaaabaGaaGOmam aavababeqcfasaaiaaicdaaKqbagqabaGaamyBaaaaaaGaamivamaa BaaajuaibaGaaG4maiabgkHiTiaaikdaaKqbagqaamaabmaabaGaam yqaiaacYcacaWGcbGaaiilaiaad6gacaGGSaGaamiBaaGaayjkaiaa wMcaaaGaayjkaiaawMcaaaaaaa@FFEA@ (43.1)

E ncD ( Θ, θ ¯ ) 2Β 2 ( 2n2m1+ 2A+ m 2 ) 2 + 2 p ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( θ T sis ( A,B,n,l )+ θ ¯ 2 m 0 T 3 ) + 2mH ( 8E ) ( 4Β 2n2m+2 s 2 1 ) 2 ( ( nm1 )! ( 2n2m+2 s 2 1 )( nm+2 s 2 1 )! )( χ T s2is ( A,B,n,l )+ σ ¯ 2 m 0 T 32 ( A,B,n,l ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGebaajuaGbeaa daqadaqaaiabfI5arjaacYcacuaH4oqCgaqeaaGaayjkaiaawMcaai abggMi6kabgkHiTmaalaaabaWaaubiaeqabeqcfasaaiaaikdaaKqb agaacaaIYaGaeuOKdieaaaqaamaavacabeqabKqbGeaacaaIYaaaju aGbaWaaeWaaeaacaaIYaGaamOBaiabgkHiTiaaikdacaWGTbGaeyOe I0IaaGymaiabgUcaRmaakaaabaGaaGOmaiaadgeacqGHRaWkcaWGTb WaaWbaaKqbGeqabaGaaGOmaaaaaKqbagqaaaGaayjkaiaawMcaaaaa aaGaey4kaSYaaSaaaeaacaaIYaGaey4dIuTaamiCamaaBaaabaGaey OeI0cabeaaaeaadaqadaqaaiabgkHiTiaaiIdacaWGfbaacaGLOaGa ayzkaaaaamaabmaabaWaaSaaaeaacaaI0aGaeuOKdieabaGaaGOmai aad6gacqGHsislcaaIYaGaamyBaiabgUcaRiaaikdacaWGZbWaaSba aeaacaaIYaaabeaacqGHsislcaaIXaaaaaGaayjkaiaawMcaamaaCa aajuaibeqaaiaaikdaaaqcfa4aaeWaaeaadaWcaaqaamaabmaabaGa amOBaiabgkHiTiaad2gacqGHsislcaaIXaaacaGLOaGaayzkaaGaai yiaaqaamaabmaabaGaaGOmaiaad6gacqGHsislcaaIYaGaamyBaiab gUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaIYaaabeaajuaGcqGHsi slcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacaWGUbGaeyOeI0IaamyB aiabgUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaIYaaajuaGbeaacq GHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaaaaiaawIcacaGLPaaa daqadaqaaiabeI7aXjaadsfadaWgaaqcfasaaiaadohacqGHsislca WGPbGaam4Caaqabaqcfa4aaeWaaeaacaWGbbGaaiilaiaadkeacaGG SaGaamOBaiaacYcacaWGSbaacaGLOaGaayzkaaGaey4kaSYaaSaaae aacuaH4oqCgaqeaaqaaiaaikdadaqfqaqabKqbGeaacaaIWaaajuaG beqaaiaad2gaaaaaaiaadsfadaWgaaqcfasaaiaaiodaaKqbagqaaa GaayjkaiaawMcaaaGcbaqcfaOaey4kaSYaaSaaaeaacaaIYaGaey4d IuTaamyBaiaadIeaaeaadaqadaqaaiabgkHiTiaaiIdacaWGfbaaca GLOaGaayzkaaaaamaabmaabaWaaSaaaeaacaaI0aGaeuOKdieabaGa aGOmaiaad6gacqGHsislcaaIYaGaamyBaiabgUcaRiaaikdacaWGZb WaaSbaaKqbGeaacaaIYaaajuaGbeaacqGHsislcaaIXaaaaaGaayjk aiaawMcaamaaCaaabeqcfasaaiaaikdaaaqcfa4aaeWaaeaadaWcaa qaamaabmaabaGaamOBaiabgkHiTiaad2gacqGHsislcaaIXaaacaGL OaGaayzkaaGaaiyiaaqaamaabmaabaGaaGOmaiaad6gacqGHsislca aIYaGaamyBaiabgUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaIYaaa beaajuaGcqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacaWGUb GaeyOeI0IaamyBaiabgUcaRiaaikdacaWGZbWaaSbaaKqbGeaacaaI YaaajuaGbeaacqGHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaaaai aawIcacaGLPaaadaqadaqaaiabeE8aJjaadsfadaWgaaqaaiaadoha juaicaaIYaqcfaOaeyOeI0IaamyAaiaadohaaeqaamaabmaabaGaam yqaiaacYcacaWGcbGaaiilaiaad6gacaGGSaGaamiBaaGaayjkaiaa wMcaaiabgUcaRmaalaaabaWaa0aaaeaacqaHdpWCaaaabaGaaGOmam aavababeqaaiaaicdaaeqabaGaamyBaaaaaaGaamivamaaBaaajuai baGaaG4maiabgkHiTiaaikdaaKqbagqaamaabmaabaGaamyqaiaacY cacaWGcbGaaiilaiaad6gacaGGSaGaamiBaaGaayjkaiaawMcaaaGa ayjkaiaawMcaaaaaaa@FEC9@ (43.2)

E ncu ( Θ, θ ¯ )2 B 2 { ( 2n ) 2 8A κ ( 2n ) 3 + 16 A 2 κ 3 ( 2n ) 3 + 48 A 2 κ 2 ( 2n ) 4 ... } α p + ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ( Θ T sis ( A,B,n,l )+ θ ¯ 2 m 0 T 3 ) mH ( 8E ) 3/2 ( 2B n ) 3 ( nl1 )! 2n( n+l )! ( χ T s3is ( A,B,n,l )+ σ ¯ 2 m 0 T 33 ( A,B,n,l ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajqwba+Faaiaad6gacaWGJbGaeyOeI0IaamyDaaqabaqc fa4aaeWaaeaacqqHyoqucaGGSaGafqiUdeNbaebaaiaawIcacaGLPa aacqGHHjIUcqGHsislcaaIYaWaaubiaeqabeqcfasaaiaaikdaaKqb agaacaWGcbaaamaacmaabaWaaubiaeqabeqcfasaaiabgkHiTiaaik daaKqbagaadaqadaqaaiaaikdacaWGUbaacaGLOaGaayzkaaaaaiab gkHiTmaalaaabaGaaGioaiaadgeaaeaacqaH6oWAaaWaaubiaeqabe qcfasaaiabgkHiTiaaiodaaKqbagaadaqadaqaaiaaikdacaWGUbaa caGLOaGaayzkaaaaaiabgUcaRmaalaaabaGaaGymaiaaiAdadaqfGa qabeqajuaibaGaaGOmaaqcfayaaiaadgeaaaaabaWaaubiaeqabeqc fasaaiaaiodaaKqbagaacqaH6oWAaaaaamaavacabeqabKqbGeaacq GHsislcaaIZaaajuaGbaWaaeWaaeaacaaIYaGaamOBaaGaayjkaiaa wMcaaaaacqGHRaWkdaWcaaqaaiaaisdacaaI4aWaaubiaeqabeqcfa saaiaaikdaaKqbagaacaWGbbaaaaqaamaavacabeqabKqbGeaacaaI YaaajuaGbaGaeqOUdSgaaaaadaqfGaqabeqajuaibaGaeyOeI0IaaG inaaqcfayaamaabmaabaGaaGOmaiaad6gaaiaawIcacaGLPaaaaaGa eyOeI0IaaiOlaiaac6cacaGGUaaacaGL7bGaayzFaaGaeyOeI0YaaS aaaeaacqaHXoqycaWGWbWaaSbaaeaacqGHRaWkaeqaaaqaamaabmaa baGaeyOeI0IaaGioaiaadweaaiaawIcacaGLPaaadaahaaqcfasabe aacaaIZaGaai4laiaaikdaaaaaaKqbaoaabmaabaWaaSaaaeaacaaI YaGaamOqaaqaaiaad6gaaaaacaGLOaGaayzkaaWaaWbaaKqbGeqaba GaaG4maaaajuaGdaWcaaqaamaabmaabaGaamOBaiabgkHiTiaadYga cqGHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaqaaiaaikdacaWGUb WaaeWaaeaacaWGUbGaey4kaSIaamiBaaGaayjkaiaawMcaaiaacgca aaWaaeWaaeaacqqHyoqucaWGubWaaSbaaKqbGeaacaWGZbGaeyOeI0 IaamyAaiaadohaaeqaaKqbaoaabmaabaGaamyqaiaacYcacaWGcbGa aiilaiaad6gacaGGSaGaamiBaaGaayjkaiaawMcaaiabgUcaRmaala aabaGafqiUdeNbaebaaeaacaaIYaWaaubeaeqajuaibaGaaGimaaqc fayabeaacaWGTbaaaaaacaWGubWaaSbaaKqbGeaacaaIZaaajuaGbe aaaiaawIcacaGLPaaaaOqaaKqbakabgkHiTmaalaaabaGaamyBaiaa dIeaaeaadaqadaqaaiabgkHiTiaaiIdacaWGfbaacaGLOaGaayzkaa WaaWbaaeqajuaibaGaaG4maiaac+cacaaIYaaaaaaajuaGdaqadaqa amaalaaabaGaaGOmaiaadkeaaeaacaWGUbaaaaGaayjkaiaawMcaam aaCaaabeqcfasaaiaaiodaaaqcfa4aaSaaaeaadaqadaqaaiaad6ga cqGHsislcaWGSbGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaacgcaae aacaaIYaGaamOBamaabmaabaGaamOBaiabgUcaRiaadYgaaiaawIca caGLPaaacaGGHaaaamaabmaabaGaeq4XdmMaamivamaaBaaabaGaam 4CaKqbGiaaiodacqGHsislcaWGPbGaam4CaaqcfayabaWaaeWaaeaa caWGbbGaaiilaiaadkeacaGGSaGaamOBaiaacYcacaWGSbaacaGLOa GaayzkaaGaey4kaSYaaSaaaeaadaqdaaqaaiabeo8aZbaaaeaacaaI YaWaaubeaeqajuaibaGaaGimaaqcfayabeaacaWGTbaaaaaacaWGub WaaSbaaKqbGeaacaaIZaGaeyOeI0IaaG4maaqcfayabaWaaeWaaeaa caWGbbGaaiilaiaadkeacaGGSaGaamOBaiaacYcacaWGSbaacaGLOa GaayzkaaaacaGLOaGaayzkaaaaaaa@F07B@  (43.3)

In this way, one can obtain the complete energy spectra for (m.i.s.) potential in (NC: 2D-RSP) and (NC: 3D-RSP) symmetries. Know the following accompanying constraint relations:

  1. The original spectrum contain two possible values of energies in ordinary two–three dimensional space which presented by equation (15),
  2. The quantum number  satisfied the interval: lm+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadYgacqGHKjYOcaWGTbGaeyizImQaey4kaSIaamiBaaaa@3E86@ , thus we have ( 2l+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdaca WGSbGaey4kaSIaaGymaaaa@39C3@ ) values for this quantum number,
  3. We have also two values for j=l+ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpcaWGSbGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3BC8@ and j=l 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpcaWGSbGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3BD3@ .

Allow us to deduce the important original results: every state in usually (two-three) dimensional space will be replace by sub-states and then the degenerated state can be take 2 i=0 n1 ( 2l+1 ) 2 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdada aeWbqaamaabmaabaGaaGOmaiaadYgacqGHRaWkcaaIXaaacaGLOaGa ayzkaaaajuaibaGaamyAaiabg2da9iaaicdaaeaacaWGUbGaeyOeI0 IaaGymaaqcfaOaeyyeIuoacqGHHjIUcaaIYaGaamOBamaaCaaajuai beqaaiaaikdaaaaaaa@48BB@ values in (NC: 2D-RSP) and (NC: 3D-RSP) symmetries . It’s clearly, that the obtained eigenvalues of energies are real and then the noncommutative diagonal Hamiltonian operators H ^ nc2ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIYaGaeyOeI0IaamyAaiaa dchaaeqaaaaa@3D0C@  and H ^ nc3ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIZaGaeyOeI0IaamyAaiaa dchaaeqaaaaa@3D0D@  are Hermitian, furthermore it’s possible to writing the two elements [ ( H ^ nc2is ) 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmisayaajaWaaSbaaKqbGeaacaWGUbGaam4yaiaaikdacqGHsisl caWGPbGaam4CaaqabaaajuaGcaGLOaGaayzkaaWaaSbaaKqbGeaaca aIXaGaaGymaaqabaaaaa@40EB@ , ( H ^ nc2is ) 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmisayaajaWaaSbaaKqbGeaacaWGUbGaam4yaiaaikdacqGHsisl caWGPbGaam4CaaqabaaajuaGcaGLOaGaayzkaaWaaSbaaKqbGeaaca aIYaGaaGOmaaqabaaaaa@40ED@  ] and [ ( H ^ nc3is ) 11 , ( H ^ nc3is ) 22 ,, ( H ^ nc3is ) 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaeWaaeaaceWGibGbaKaadaWgaaqcfasaaiaad6gacaWGJbGaaG4m aiabgkHiTiaadMgacaWGZbaajuaGbeaaaiaawIcacaGLPaaadaWgaa qcfasaaiaaigdacaaIXaaabeaajuaGcaGGSaWaaeWaaeaaceWGibGb aKaadaWgaaqcfasaaiaad6gacaWGJbGaaG4maiabgkHiTiaadMgaca WGZbaajuaGbeaaaiaawIcacaGLPaaadaWgaaqcfasaaiaaikdacaaI YaaajuaGbeaacaGGSaGaaiilamaabmaabaGabmisayaajaWaaSbaaK qbGeaacaWGUbGaam4yaiaaiodacqGHsislcaWGPbGaam4Caaqcfaya baaacaGLOaGaayzkaaWaaSbaaKqbGeaacaaIZaGaaG4maaqcfayaba aacaGLBbGaayzxaaaaaa@5B84@ , as follows, respectively:

( H ^ nc2is ) 11 = 1 2 m 0 ( 1 r r ( r r )+ 1 r 2 2 ϕ 2 )+ A r 2 B r + p + { θ ¯ 2 m 0 +( A r 4 B 2 r 3 )θ }+{ σ ¯ 2 m 0 +χ( A r 4 B 2 r 3 ) } H L      ( H ^ nc2is ) 22 = 1 2 m 0 ( 1 r r ( r r )+ 1 r 2 2 ϕ 2 )+ A r 2 B r + p { θ ¯ 2 m 0 +( A r 4 B 2 r 3 )θ }+{ σ ¯ 2 m 0 +χ( A r 4 B 2 r 3 ) } H L      MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aae WaaeaaceWGibGbaKaadaWgaaqcfasaaiaad6gacaWGJbGaaGOmaiab gkHiTiaadMgacaWGZbaabeaaaKqbakaawIcacaGLPaaadaWgaaqcfa saaiaaigdacaaIXaaabeaajuaGcqGH9aqpcqGHsisldaWcaaqaaiaa igdaaeaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqcfayabaaaam aabmaabaWaaSqaaeaacaaIXaaabaGaamOCaaaadaWcbaqaaiabgkGi 2cqaaiabgkGi2kaadkhaaaWaaeWaaeaacaWGYbWaaSqaaeaacqGHci ITaeaacqGHciITcaWGYbaaaaGaayjkaiaawMcaaiabgUcaRmaaleaa baGaaGymaaqaaiaadkhadaahaaqcfasabeaacaaIYaaaaaaajuaGda WcbaqaaiabgkGi2oaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaeyOa IyRaeqy1dy2aaWbaaeqajuaibaGaaGOmaaaaaaaajuaGcaGLOaGaay zkaaGaey4kaSYaaSaaaeaacaWGbbaabaGaamOCamaaCaaajuaibeqa aiaaikdaaaaaaKqbakabgkHiTmaalaaabaGaamOqaaqaaiaadkhaaa Gaey4kaSIaamiCamaaBaaabaGaey4kaScabeaadaGadaqaamaalaaa baWaa0aaaeaacqaH4oqCaaaabaGaaGOmaiaad2gadaWgaaqcfasaai aaicdaaKqbagqaaaaacqGHRaWkdaqadaqaamaalaaabaGaamyqaaqa aiaadkhadaahaaqcfasabeaacaaI0aaaaaaajuaGcqGHsisldaWcaa qaaiaadkeaaeaacaaIYaGaamOCamaaCaaajuaibeqaaiaaiodaaaaa aaqcfaOaayjkaiaawMcaaiabeI7aXbGaay5Eaiaaw2haaiabgUcaRm aacmaabaWaaSaaaeaadaqdaaqaaiabeo8aZbaaaeaacaaIYaGaamyB amaaBaaajuaibaGaaGimaaqabaaaaKqbakabgUcaRiabeE8aJnaabm aabaWaaSaaaeaacaWGbbaabaGaamOCamaaCaaajuaibeqaaiaaisda aaaaaKqbakabgkHiTmaalaaabaGaamOqaaqaaiaaikdacaWGYbWaaW baaKqbGeqabaGaaG4maaaaaaaajuaGcaGLOaGaayzkaaaacaGL7bGa ayzFaaGaaGPaVpaaFiaabaGaamisaaGaay51GaWaa8HaaeaacaWGmb aacaGLxdcacaqGGaGaaeiiaiaabccacaqGGaaakeaajuaGdaqadaqa aiqadIeagaqcamaaBaaajuaibaGaamOBaiaadogacaaIYaGaeyOeI0 IaamyAaiaadohaaeqaaaqcfaOaayjkaiaawMcaamaaBaaajuaibaGa aGOmaiaaikdaaKqbagqaaiabg2da9iabgkHiTmaalaaabaGaaGymaa qaaiaaikdacaWGTbWaaSbaaKqbGeaacaaIWaaabeaaaaqcfa4aaeWa aeaadaWcbaqaaiaaigdaaeaacaWGYbaaamaaleaabaGaeyOaIylaba GaeyOaIyRaamOCaaaadaqadaqaaiaadkhadaWcbaqaaiabgkGi2cqa aiabgkGi2kaadkhaaaaacaGLOaGaayzkaaGaey4kaSYaaSqaaeaaca aIXaaabaGaamOCamaaCaaabeqcfasaaiaaikdaaaaaaKqbaoaaleaa baGaeyOaIy7aaWbaaKqbGeqabaGaaGOmaaaaaKqbagaacqGHciITcq aHvpGzdaahaaqcfasabeaacaaIYaaaaaaaaKqbakaawIcacaGLPaaa cqGHRaWkdaWcaaqaaiaadgeaaeaacaWGYbWaaWbaaKqbGeqabaGaaG OmaaaaaaqcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaamOCaaaacqGH RaWkcaWGWbWaaSbaaeaacqGHsislaeqaamaacmaabaWaaSaaaeaada qdaaqaaiabeI7aXbaaaeaacaaIYaGaamyBamaaBaaajuaibaGaaGim aaqcfayabaaaaiabgUcaRmaabmaabaWaaSaaaeaacaWGbbaabaGaam OCamaaCaaajuaibeqaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGa amOqaaqaaiaaikdacaWGYbWaaWbaaKqbGeqabaGaaG4maaaaaaaaju aGcaGLOaGaayzkaaGaeqiUdehacaGL7bGaayzFaaGaey4kaSYaaiWa aeaadaWcaaqaamaanaaabaGaeq4WdmhaaaqaaiaaikdacaWGTbWaaS baaKqbGeaacaaIWaaabeaaaaqcfaOaey4kaSIaeq4Xdm2aaeWaaeaa daWcaaqaaiaadgeaaeaacaWGYbWaaWbaaKqbGeqabaGaaGinaaaaaa qcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaaGOmaiaadkhadaahaaqc fasabeaacaaIZaaaaaaaaKqbakaawIcacaGLPaaaaiaawUhacaGL9b aacaaMc8+aa8HaaeaacaWGibaacaGLxdcadaWhcaqaaiaadYeaaiaa wEniaiaabccacaqGGaGaaeiiaiaabccaaaaa@03BD@ (44.1)
and
{ ( H ^ nc3is ) 11 = 1 2 m 0 [ 1 r 2 r ( r 2 r )+ 1 r 2 sinθ θ ( sinθ θ )+ 1 r 2 ( sinθ ) 2 2 ϕ 2 ]+ A r 2 B r + p + [ Θ( A r 4 B 2 r 3 )+ θ ¯ 2 m 0 ]+{ σ ¯ 2 m 0 +χ( A r 4 B 2 r 3 ) } H L     for  j=+1/2  spin up  ( H ^ nc3is ) 22 = 1 2 m 0 [ 1 r 2 r ( r 2 r )+ 1 r < sinθ θ ( sinθ θ )+ 1 r 2 ( sinθ ) 2 2 ϕ 2 ]+ A r 2 B r p [ Θ( A r 4 B 2 r 3 )+ θ ¯ 2 m 0 ]+{ σ ¯ 2 m 0 +χ( A r 4 B 2 r 3 ) } H L     for  j=1/2  spin down  ( H ^ nc3is ) 33 = 1 2 m 0 [ 1 r 2 r ( r 2 r )+ 1 r 2 sinθ θ ( sinθ θ )+ 1 r 2 ( sinθ ) 2 2 ϕ 2 ]+ A r 2 B r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaea qabeaadaqadaqaaiqadIeagaqcamaaBaaajuaibaGaamOBaiaadoga caaIZaGaeyOeI0IaamyAaiaadohaaKqbagqaaaGaayjkaiaawMcaam aaBaaajuaibaGaaGymaiaaigdaaeqaaKqbakabg2da9iabgkHiTmaa laaabaGaaGymaaqaaiaaikdacaWGTbWaaSbaaKqbGeaacaaIWaaaju aGbeaaaaWaamWaaeaadaWcaaqaaiaaigdaaeaadaqfGaqabeqajuai baGaaGOmaaqcfayaaiaadkhaaaaaamaalaaabaGaeyOaIylabaGaey OaIyRaamOCaaaadaqadaqaamaavacabeqabKqbGeaacaaIYaaajuaG baGaamOCaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadkhaaaaaca GLOaGaayzkaaGaey4kaSYaaSaaaeaacaaIXaaabaWaaubiaeqabeqc fasaaiaaikdaaKqbagaacaWGYbaaaiGacohacaGGPbGaaiOBaiabeI 7aXbaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabeI7aXbaadaqadaqa aiGacohacaGGPbGaaiOBaiabeI7aXnaalaaabaGaeyOaIylabaGaey OaIyRaeqiUdehaaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGaaGym aaqaamaavacabeqabKqbGeaacaaIYaaajuaGbaGaamOCaaaadaqfGa qabeqajuaibaGaaGOmaaqcfayaamaabmaabaGaci4CaiaacMgacaGG UbGaeqiUdehacaGLOaGaayzkaaaaaaaadaWcaaqaamaavacabeqabK qbGeaacaaIYaaajuaGbaGaeyOaIylaaaqaaiabgkGi2oaavacabeqa bKqbGeaacaaIYaaajuaGbaGaeqy1dygaaaaaaiaawUfacaGLDbaacq GHRaWkdaWcaaqaaiaadgeaaeaacaWGYbWaaWbaaKqbGeqabaGaaGOm aaaaaaqcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaamOCaaaacqGHRa WkaeaadaqfqaqabeaacqGHRaWkaeqabaGaaeiCaaaadaWadaqaaiab fI5arnaabmaabaWaaSaaaeaacaWGbbaabaWaaubiaeqabeqcfasaai aaisdaaKqbagaacaWGYbaaaaaacqGHsisldaWcaaqaaiaadkeaaeaa caaIYaWaaubiaeqabeqcfasaaiaaiodaaKqbagaacaWGYbaaaaaaai aawIcacaGLPaaacqGHRaWkdaWcaaqaaiqbeI7aXzaaraaabaGaaGOm amaavababeqcfasaaiaaicdaaKqbagqabaGaamyBaaaaaaaacaGLBb GaayzxaaGaey4kaSYaaiWaaeaadaWcaaqaamaanaaabaGaeq4Wdmha aaqaaiaaikdacaWGTbWaaSbaaKqbGeaacaaIWaaajuaGbeaaaaGaey 4kaSIaeq4Xdm2aaeWaaeaadaWcaaqaaiaadgeaaeaacaWGYbWaaWba aeqajuaibaGaaGinaaaaaaqcfaOaeyOeI0YaaSaaaeaacaWGcbaaba GaaGOmaiaadkhadaahaaqabKqbGeaacaaIZaaaaaaaaKqbakaawIca caGLPaaaaiaawUhacaGL9baacaaMc8+aa8HaaeaacaWGibaacaGLxd cadaWhcaqaaiaadYeaaiaawEniaiaabccacaqGGaGaaeiiaiaabcca caqGMbGaae4BaiaabkhacaqGGaGaaeiiaiaabQgacqGH9aqpcqWIte cBcqGHRaWkcaqGXaGaae4laiaabkdacaqGGaGaaeiiaiabgkDiElaa bohacaqGWbGaaeyAaiaab6gacaqGGaGaaeyDaiaabchacaqGGaaaba WaaeWaaeaaceWGibGbaKaadaWgaaqcfasaaiaad6gacaWGJbGaaG4m aiabgkHiTiaadMgacaWGZbaajuaGbeaaaiaawIcacaGLPaaadaWgaa qcfasaaiaaikdacaaIYaaajuaGbeaacqGH9aqpcqGHsisldaWcaaqa aiaaigdaaeaacaaIYaGaamyBamaaBaaajuaibaGaaGimaaqabaaaaK qbaoaadmaabaWaaSaaaeaacaaIXaaabaWaaubiaeqabeqcfasaaiaa ikdaaKqbagaacaWGYbaaaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2k aadkhaaaWaaeWaaeaadaqfGaqabeqajuaibaGaaGOmaaqcfayaaiaa dkhaaaWaaSaaaeaacqGHciITaeaacqGHciITcaWGYbaaaaGaayjkai aawMcaaiabgUcaRmaalaaabaGaaGymaaqaamaavacabeqabKqbafaa cqGH8aapaKqbagaacaWGYbaaaiGacohacaGGPbGaaiOBaiabeI7aXb aadaWcaaqaaiabgkGi2cqaaiabgkGi2kabeI7aXbaadaqadaqaaiGa cohacaGGPbGaaiOBaiabeI7aXnaalaaabaGaeyOaIylabaGaeyOaIy RaeqiUdehaaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGaaGymaaqa amaavacabeqabKqbGeaacaaIYaaajuaGbaGaamOCaaaadaqfGaqabe qajuaibaGaaGOmaaqcfayaamaabmaabaGaci4CaiaacMgacaGGUbGa eqiUdehacaGLOaGaayzkaaaaaaaadaWcaaqaamaavacabeqabKqbGe aacaaIYaaajuaGbaGaeyOaIylaaaqaaiabgkGi2oaavacabeqabKqb GeaacaaIYaaajuaGbaGaeqy1dygaaaaaaiaawUfacaGLDbaacqGHRa WkdaWcaaqaaiaadgeaaeaacaWGYbWaaWbaaeqajuaibaGaaGOmaaaa aaqcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaamOCaaaaaeaadaqfqa qabeaacqGHsislaeqabaGaaeiCaaaadaWadaqaaiabfI5arnaabmaa baWaaSaaaeaacaWGbbaabaWaaubiaeqabeqcfasaaiaaisdaaKqbag aacaWGYbaaaaaacqGHsisldaWcaaqaaiaadkeaaeaacaaIYaWaaubi aeqabeqcfasaaiaaiodaaKqbagaacaWGYbaaaaaaaiaawIcacaGLPa aacqGHRaWkdaWcaaqaaiqbeI7aXzaaraaabaGaaGOmamaavababeqc fasaaiaaicdaaKqbagqabaGaamyBaaaaaaaacaGLBbGaayzxaaGaey 4kaSYaaiWaaeaadaWcaaqaamaanaaabaGaeq4Wdmhaaaqaaiaaikda caWGTbWaaSbaaKqbGeaacaaIWaaabeaaaaqcfaOaey4kaSIaeq4Xdm 2aaeWaaeaadaWcaaqaaiaadgeaaeaacaWGYbWaaWbaaKqbGeqabaGa aGinaaaaaaqcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaaGOmaiaadk hadaahaaqcfasabeaacaaIZaaaaaaaaKqbakaawIcacaGLPaaaaiaa wUhacaGL9baacaaMc8+aa8HaaeaacaWGibaacaGLxdcadaWhcaqaai aadYeaaiaawEniaiaabccacaqGGaGaaeiiaiaabccacaqGMbGaae4B aiaabkhacaqGGaGaaeiiaiaabQgacqGH9aqpcqWItecBcqGHsislca qGXaGaae4laiaabkdacaqGGaGaaeiiaiabgkDiElaabohacaqGWbGa aeyAaiaab6gacaqGGaGaaeizaiaab+gacaqG3bGaaeOBaiaabccaae aadaqadaqaaiqadIeagaqcamaaBaaajuaibaGaamOBaiaadogacaaI ZaGaeyOeI0IaamyAaiaadohaaKqbagqaaaGaayjkaiaawMcaamaaBa aajuaibaGaaG4maiaaiodaaeqaaKqbakabg2da9iabgkHiTmaalaaa baGaaGymaaqaaiaaikdacaWGTbWaaSbaaKqbGeaacaaIWaaajuaGbe aaaaWaamWaaeaadaWcaaqaaiaaigdaaeaadaqfGaqabeqajuaibaGa aGOmaaqcfayaaiaadkhaaaaaamaalaaabaGaeyOaIylabaGaeyOaIy RaamOCaaaadaqadaqaamaavacabeqabKqbGeaacaaIYaaajuaGbaGa amOCaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadkhaaaaacaGLOa GaayzkaaGaey4kaSYaaSaaaeaacaaIXaaabaWaaubiaeqabeqcfasa aiaaikdaaKqbagaacaWGYbaaaiGacohacaGGPbGaaiOBaiabeI7aXb aadaWcaaqaaiabgkGi2cqaaiabgkGi2kabeI7aXbaadaqadaqaaiGa cohacaGGPbGaaiOBaiabeI7aXnaalaaabaGaeyOaIylabaGaeyOaIy RaeqiUdehaaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGaaGymaaqa amaavacabeqabKqbGeaacaaIYaaajuaGbaGaamOCaaaadaqfGaqabe qajuaibaGaaGOmaaqcfayaamaabmaabaGaci4CaiaacMgacaGGUbGa eqiUdehacaGLOaGaayzkaaaaaaaadaWcaaqaamaavacabeqabKqbGe aacaaIYaaajuaGbaGaeyOaIylaaaqaaiabgkGi2oaavacabeqabKqb GeaacaaIYaaajuaGbaGaeqy1dygaaaaaaiaawUfacaGLDbaacqGHRa WkdaWcaaqaaiaadgeaaeaacaWGYbWaaWbaaeqajuaibaGaaGOmaaaa aaqcfaOaeyOeI0YaaSaaaeaacaWGcbaabaGaamOCaaaaaaGaay5Eaa aaaa@CCD6@ (44.2)

On the other hand, the above obtain results (44.1) and (44.2) allow us to constructing the diagonal anisotropic matrixes [ ( H ^ nc2is ) 11 ( H ^ nc2is ) 22 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaeWaaeaaceWGibGbaKaadaWgaaqcfasaaiaad6gacaWGJbGaaGOm aiabgkHiTiaadMgacaWGZbaajuaGbeaaaiaawIcacaGLPaaadaWgaa qcfasaaiaaigdacaaIXaaabeaajuaGcqGHGjsUdaqadaqaaiqadIea gaqcamaaBaaajuaibaGaamOBaiaadogacaaIYaGaeyOeI0IaamyAai aadohaaeqaaaqcfaOaayjkaiaawMcaamaaBaaajuaibaGaaGOmaiaa ikdaaeqaaaqcfaOaay5waiaaw2faaaaa@5034@  and [ ( H ^ nc3is ) 11 ( H ^ nc3is ) 22 ] ( H ^ nc3is ) 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaeWaaeaaceWGibGbaKaadaWgaaqcfasaaiaad6gacaWGJbGaaG4m aiabgkHiTiaadMgacaWGZbaabeaaaKqbakaawIcacaGLPaaadaWgaa qcfasaaiaaigdacaaIXaaabeaajuaGcqGHGjsUdaqadaqaaiqadIea gaqcamaaBaaajuaibaGaamOBaiaadogacaaIZaGaeyOeI0IaamyAai aadohaaKqbagqaaaGaayjkaiaawMcaamaaBaaajuaibaGaaGOmaiaa ikdaaKqbagqaaaGaay5waiaaw2faaiabgcMi5oaabmaabaGabmisay aajaWaaSbaaKqbGeaacaWGUbGaam4yaiaaiodacqGHsislcaWGPbGa am4CaaqcfayabaaacaGLOaGaayzkaaWaaSbaaKqbGeaacaaIYaGaaG Omaaqabaaaaa@5C72@ of the Hamiltonian operators H ^ nc2ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIYaGaeyOeI0IaamyAaiaa dchaaKqbagqaaaaa@3D9A@  and H ^ nc3ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIZaGaeyOeI0IaamyAaiaa dchaaeqaaaaa@3D0D@  for (m.i.s.) potential in (NC: 2D-RSP) and (NC: 3D-RSP) symmetries respectively, as:
H ^ nc2ip =( ( H ^ nc2is ) 11 0 0 ( H nc2is ) 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIYaGaeyOeI0IaamyAaiaa dchaaeqaaKqbakabg2da9maabmaabaqbaeqabiGaaaqaamaabmaaba GabmisayaajaWaaSbaaKqbGeaacaWGUbGaam4yaiaaikdacqGHsisl caWGPbGaam4CaaqabaaajuaGcaGLOaGaayzkaaWaaSbaaKqbGeaaca aIXaGaaGymaaqcfayabaaabaGaaGimaaqaaiaaicdaaeaadaqadaqa aiaadIeadaWgaaqcfasaaiaad6gacaWGJbGaaGOmaiabgkHiTiaadM gacaWGZbaabeaaaKqbakaawIcacaGLPaaadaWgaaqcfasaaiaaikda caaIYaaajuaGbeaaaaaacaGLOaGaayzkaaaaaa@579F@ (45.1)
and
H ^ nc3is =( ( H nc3is ) 11 0 0 0 ( H nc3is ) 22 0 0 0 ( H nc3is ) 33 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacaaIZaGaeyOeI0IaamyAaiaa dohaaKqbagqaaiabg2da9maabmaabaqbaeqabmWaaaqaamaabmaaba GaamisamaaBaaajuaibaGaamOBaiaadogacaaIZaGaeyOeI0IaamyA aiaadohaaKqbagqaaaGaayjkaiaawMcaamaaBaaajuaibaGaaGymai aaigdaaeqaaaqcfayaaiaaicdaaeaacaaIWaaabaGaaGimaaqaamaa bmaabaGaamisamaaBaaajuaibaGaamOBaiaadogacaaIZaGaeyOeI0 IaamyAaiaadohaaKqbagqaaaGaayjkaiaawMcaamaaBaaajuaibaGa aGOmaiaaikdaaKqbagqaaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaamaabmaabaGaamisamaaBaaajuaibaGaamOBaiaadogacaaIZaGa eyOeI0IaamyAaiaadohaaeqaaaqcfaOaayjkaiaawMcaamaaBaaaju aibaGaaG4maiaaiodaaKqbagqaaaaaaiaawIcacaGLPaaaaaa@6579@ (45.2)
Which allows us to obtain the original results for this investigation: the obtained Hamiltonian operators (45.1) and (45.2) can be describing atom which has two permanent dipoles: the first is electric dipole moment and the second is magnetic moment in external stationary electromagnetic field. It is important to notice that, the appearance of the polarization states of a fermionic particle for (m.i.s.) potential indicate to the validity of obtained results at very high energy where the two relativistic equations: Klein-Gordon and Dirac will be applied, which allowing to apply these results of various Nano-particles at Nano scales.

Conclusion

In this study we have performed the exact analytical bound state solutions: the energy spectra and the corresponding noncommutative Hamiltonians for the two and three dimensional Schrödinger equations in polar and spherical coordinates for modified inverse-squire potential by using generalization Boopp’s Shift method and standard perturbation theory. It is found that the energy eigenvalues depend on the dimensionality of the problem and new atomic quantum numbers ( j=l±1/1,s=±1/2,l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpcaWGSbGaeyySaeRaaGymaiaac+cacaaIXaGaaiilaiaadoha cqGH9aqpcqGHXcqScaaIXaGaai4laiaaikdacaGGSaGaamiBaaaa@45DD@ and the angular momentum quantum number in addition to two infinitesimals parameters ( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ , θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaraaaaa@3847@ ) and ( Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI5arb aa@37F0@ , θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaraaaaa@3847@ ) in the symmetries of (NC: 2D-RSP) and (NC: 3D-RSP). And we also showed that the obtained energy spectra degenerate and every old state will be replaced by 2( 2l+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdada qadaqaaiaaikdacaWGSbGaey4kaSIaaGymaaGaayjkaiaawMcaaaaa @3C08@ sub-states. Finally, we expect that the results of our research are valid in the high energies, thus the (m.s.e) can gives the same results of Dirac and Klein-Gordon equations.

Acknowledgement

This work was supported with search laboratory of: Physique et Chimie des matériaux, in university of M'sila, Algeria.

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